A fundamental problem in data management is to find the elements in an array that match a query. Recently, learned indexes are being extensively used to solve this problem, where they learn a model to predict the location of the items in the array. They are empirically shown to outperform non-learned methods (e.g., B-trees or binary search that answer queries in $O(\log n)$ time) by orders of magnitude. However, success of learned indexes has not been theoretically justified. Only existing attempt shows the same query time of $O(\log n)$, but with a constant factor improvement in space complexity over non-learned methods, under some assumptions on data distribution. In this paper, we significantly strengthen this result, showing that under mild assumptions on data distribution, and the same space complexity as non-learned methods, learned indexes can answer queries in $O(\log\log n)$ expected query time. We also show that allowing for slightly larger but still near-linear space overhead, a learned index can achieve $O(1)$ expected query time. Our results theoretically prove learned indexes are orders of magnitude faster than non-learned methods, theoretically grounding their empirical success.

本文从理论上证明，在数据分布的温和假设下，具有与非学习方法相同空间复杂度的学习索引可以在期望的 O(loglog n) 查询时间内回答查询，从而进一步巩固了学习索引的实证成功。

关于学习索引的分布依赖子对数查询时间