Asymptotically Optimal Learning for Parametric Prophet Inequalities

2026-06-25

AI Summary

This work addresses the prophet inequality problem under reward distributions belonging to exponential parameter families—such as exponential, Pareto, and bounded-support power-law distributions—and investigates online learning methods that achieve asymptotically optimal competitive ratios using only online observations. By exploiting the explicit parametric structure of these distributions, the authors devise a confidence interval–based dynamic programming strategy that approaches full-information optimal performance without requiring offline samples. The main contributions include establishing, for the first time, the theoretical asymptotic limits of optimal competitive ratios: in the unbounded-support case, the ratio attains $(\theta/(\theta - c_+))^{c_+/\theta} / \Gamma(1 - c_+/\theta)$, while in the bounded-support setting, it converges to 1. Furthermore, the paper proposes the first online-learning algorithm that achieves this limit using solely online data, with both theoretical analysis and experiments confirming its efficacy.

Abstract

We study learning in prophet inequalities with i.i.d. rewards drawn from an exponential-type parametric family with an unknown parameter $θ$, a class that includes exponential, Pareto, and bounded-support power-family distributions. We first characterize the optimal full-information asymptotic competitive ratio for this family. In the unbounded-support case, the limit is $ {\left(θ/({θ-c_+})\right)^{c_+/θ}}/ {Γ(1-c_+/θ)},$ while in the bounded-support case, the limit is $1$. We then propose a confidence-based dynamic-programming policy for online learning. By exploiting the explicit parametric structure, the policy achieves the same optimal asymptotic competitive ratio using only online observations, without external offline samples. We further derive distribution-specific convergence rates for canonical examples. Finally, numerical experiments on synthetic instances illustrate the performance of our algorithm.

Problem

• prophet inequalities
• parametric family
• asymptotic competitive ratio
• online learning
• exponential-type distributions

Innovation

• prophet inequalities
• parametric family
• asymptotic competitive ratio
• online learning
• dynamic programming