Threshold Testing and Semi-Online Prophet Inequalities

TL;DR

This paper investigates threshold testing algorithms for selecting large values among i.i.d. variables, demonstrating how adaptive strategies outperform non-adaptive ones and approaching optimal ratios with multiple tests.

Contribution

It introduces adaptive algorithms that significantly improve the competitive ratio in threshold testing, surpassing the standard prophet inequality bounds.

Findings

Adaptive algorithms achieve at least 0.869 ratio.

Non-adaptive algorithms are limited to about 0.745 ratio.

Multiple testing allows near-optimal performance approaching ratio 1.

Abstract

We study threshold testing, an elementary probing model with the goal to choose a large value out of $n$ i.i.d. random variables. An algorithm can test each variable $X_i$ once for some threshold $t_i$, and the test returns binary feedback whether $X_i \ge t_i$ or not. Thresholds can be chosen adaptively or non-adaptively by the algorithm. Given the results for the tests of each variable, we then select the variable with highest conditional expectation. We compare the expected value obtained by the testing algorithm with expected maximum of the variables. Threshold testing is a semi-online variant of the gambler's problem and prophet inequalities. Indeed, the optimal performance of non-adaptive algorithms for threshold testing is governed by the standard i.i.d. prophet inequality of approximately $0.745+o(1)$ as $n \to \infty$. We show how adaptive algorithms can significantly improve upon this ratio. Our adaptive testing strategy guarantees a competitive ratio of at least $0.869-o(1)$. Moreover, we show that there are distributions that admit only a constant ratio $c < 1$, even when $n \to \infty$. Finally, when each box can be tested multiple times (with $n$ tests in total), we design an algorithm that achieves a ratio of $1-o(1)$.