Sample-Based Matroid Prophet Inequalities

We study matroid prophet inequalities when distributions are unknown and accessible only through samples. While single-sample prophet inequalities for special matroids are known, no constant-factor competitive algorithm with even a sublinear number of samples was known for general matroids. Adding more to the stake, the single-sample version of the question for general matroids has close (two-way) connections with the long-standing matroid secretary conjecture. In this work, we give a (14 - ε)-competitive matroid prophet inequality with only O_ε(poly log n) samples. Our algorithm consists of two parts: (i) a novel quantile-based reduction from matroid prophet inequalities to online contention resolution schemes (OCRSs) with O_ε(log n) samples, and (ii) a (14 - ε)-selectable matroid OCRS with O_ε(poly log n) samples which carefully addresses an adaptivity challenge.