Learned indexes leverage machine learning models to accelerate query answering in databases, showing impressive practical performance. However, theoretical understanding of these methods remains incomplete. Existing research suggests that learned indexes have superior asymptotic complexity compared to their non-learned counterparts, but these findings have been established under restrictive probabilistic assumptions. Specifically, for a sorted array with $n$ elements, it has been shown that learned indexes can find a key in $O(\log(\log n))$ expected time using at most linear space, compared with $O(\log n)$ for non-learned methods. In this work, we prove $O(1)$ expected time can be achieved with at most linear space, thereby establishing the tightest upper bound so far for the time complexity of an asymptotically optimal learned index. Notably, we use weaker probabilistic assumptions than prior work, meaning our results generalize previous efforts. Furthermore, we introduce a new measure of statistical complexity for data. This metric exhibits an information-theoretical interpretation and can be estimated in practice. This characterization provides further theoretical understanding of learned indexes, by helping to explain why some datasets seem to be particularly challenging for these methods.

翻译：学习索引利用机器学习模型加速数据库中的查询响应，展现出令人瞩目的实际性能。然而，对这些方法的理论理解仍不完善。现有研究表明，与非学习型索引相比，学习索引具有更优的渐近复杂度，但这些结论是在严格概率假设下建立的。具体而言，对于包含$n$个元素的有序数组，已有工作证明学习索引能在期望时间$O(\log(\log n))$内定位键值，且最多使用线性空间，而非学习方法则需要$O(\log n)$时间。本文证明，在最多使用线性空间的条件下，期望时间可达到$O(1)$，从而为渐近最优学习索引的时间复杂度建立了当前最严格的上界。值得注意的是，本文采用了比以往工作更弱的概率假设，这意味着我们的结论推广了先前的研究成果。此外，我们引入了一种新的数据统计复杂度度量。该度量具有信息论解释，并可在实践中估计。这一刻画通过帮助解释为何某些数据集对学习索引方法特别具有挑战性，进一步深化了对学习索引的理论理解。