Learning in Prophet Inequalities with Noisy Observations

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.

We study repeated prophet inequalities under prefix feedback. In each of $T$ rounds, a learner encounters fresh values drawn independently from $n$ boxes with unknown $[0,1]$-supported distributions in a fixed order and must irrevocably accept one, observing only the prefix up to its stopping box. Regret is measured against the optimal stopping policy that knows the distributions. We give an efficient algorithm achieving $\widetilde O(\sqrt{T})$ expected regret, matching the lower bound up to logarithmic factors. Our algorithm explores directly through near-optimal policies, combining empirical backward induction with box-specific reach bonuses. A relative-drop aggregation rule then exploits the nesting structure of observed prefixes to preserve exploration, thereby removing the polynomial dependence on the box number $n$. This resolves an open question posed by Liu et al. (2025).

Stochastic optimization is a widely used approach for optimization under uncertainty, where uncertain input parameters are modeled by random variables. Exact or approximation algorithms have been obtained for several fundamental problems in this area. However, a significant limitation of this approach is that it requires full knowledge of the underlying probability distributions. Can we still get good (approximation) algorithms if these distributions are unknown, and the algorithm needs to learn them through repeated interactions? In this paper, we resolve this question for a large class of ''monotone'' stochastic problems, by providing a generic online learning algorithm with $\sqrt{T\log(T)}$ regret relative to the best approximation algorithm (under known distributions). Importantly, our online algorithm works in a semi-bandit setting, where in each period, the algorithm only observes samples from the random variables that were actually probed. Moreover, our result extends to settings with censored and binary feedback, where the policy only observes truncated or thresholded versions of the probed variables. Our framework applies to several fundamental problems such as prophet inequality, Pandora's box, stochastic knapsack, single-resource revenue management and sequential posted pricing.