Optimal Stopping with a Predicted Prior

TL;DR

This work introduces optimal stopping with a predicted prior, a model that sits between the secretary problem and prophet inequalities by providing a machine-learned prior that may be misspecified. The authors design bi-criteria algorithms that simultaneously leverage the predicted prior when accurate and retain worst-case performance guarantees when it is not, achieving improved consistency-robustness trade-offs for MaxExp and MaxProb. A novel implicit sharding technique reduces analysis to the asymptotic regime, complemented by tractable factor-revealing linear programs that bound what any algorithm can achieve and enable a hardness result for MaxProb. The key contributions include concrete bi-criteria constructions with explicit consistency-robustness characterizations, an asymptotic analysis yielding sharp limits, and a formal hardness result showing that no algorithm can be simultaneously optimal in both metrics for MaxProb, establishing a clear frontier for this problem family.

Abstract

There are two major models of value uncertainty in the optimal stopping literature: the secretary model, which assumes no prior knowledge, and the prophet inequality model, which assumes full information about value distributions. In practice, decision makers often rely on machine-learned priors that may be erroneous. Motivated by this gap, we formulate the model of optimal stopping with a predicted prior to design algorithms that are both consistent, exploiting the prediction when accurate, and robust, retaining worst-case guarantees when it is not. Existing secretary and prophet inequality algorithms are either pessimistic in consistency or not robust to misprediction. A randomized combination only interpolates their guarantees linearly. We show that a family of bi-criteria algorithms achieves improved consistency-robustness trade-offs, both for maximizing the expected accepted value and for maximizing the probability of accepting the maximum value. We further prove that for the latter objective, no algorithm can simultaneously match the best prophet inequality algorithm in consistency, and the best secretary algorithm in robustness.

Figures (3)

• Figure 1: Illustration of our results and a comparison with the baselines
• Figure 2: Illustration of the robust threshold function corresponding to Gilbert–Mosteller algorithm, with $n=10$, $\beta = 1/3$, $\lambda_1= \mathrm{e}^{\mathrm{e}^{\mathcal{W}_{-1}(-\beta)}(-1/3)}\approx 0.220$ and $\lambda_2= \mathrm{e}^{\mathrm{e}^{\mathcal{W}_{0}(-\beta)}(-1/3)}\approx 0.538$.
• Figure 3: Example solution of $\theta$ when $\beta=0.01$, $m=300$, and $\alpha\approx 0.6908$.

Theorems & Definitions (58)

• theorem 1: MaxExp algorithm
• theorem 2: MaxProb algorithm
• theorem 3: MaxProb hardness, informal version of \ref{['thm:hardness-maxprob']}
• definition 1: Consistency
• definition 2: Robustness
• definition 3
• definition 4
• proposition 4
• proof
• corollary 5