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 $(\frac14 - \varepsilon)$-competitive matroid prophet inequality with only $O_\varepsilon(\mathrm{poly} \log n)$ samples. Our algorithm consists of two parts: (i) a novel quantile-based reduction from matroid prophet inequalities to online contention resolution schemes (OCRSs) with $O_\varepsilon(\log n)$ samples, and (ii) a $(\frac14 - \varepsilon)$-selectable matroid OCRS with $O_\varepsilon(\mathrm{poly} \log n)$ samples which carefully addresses an adaptivity challenge.

翻译：本研究探讨了在分布未知且仅能通过样本获取的情况下，拟阵先知不等式问题。尽管针对特殊拟阵的单样本先知不等式已有研究，但对于一般拟阵，即使使用亚线性数量的样本，此前也未知具有常数竞争比的算法。更重要的是，一般拟阵的单样本版本问题与长期存在的拟阵秘书猜想存在密切（双向）联系。本文中，我们提出了一个仅需$O_\varepsilon(\mathrm{poly} \log n)$个样本即可实现的$(\frac14 - \varepsilon)$竞争拟阵先知不等式算法。该算法包含两部分：(i) 一种新颖的基于分位数的归约方法，使用$O_\varepsilon(\log n)$个样本将拟阵先知不等式问题转化为在线竞争消解方案（OCRS）问题；(ii) 一个具有$O_\varepsilon(\mathrm{poly} \log n)$个样本的$(\frac14 - \varepsilon)$可选拟阵OCRS，该方案通过精细处理自适应性挑战实现。