Why Are Learned Indexes So Effective but Sometimes Ineffective?

TL;DR

This paper analyzes the theoretical efficiency and practical limitations of learned indexes, especially PGM-Index, and proposes PGM++ to significantly improve lookup performance while maintaining space efficiency.

Contribution

It provides the tightest theoretical bounds for PGM-Index and introduces PGM++, a new extension that enhances practical performance through hybrid search strategies.

Findings

PGM-Index can achieve O(log log N) lookup time with high probability.

Querying PGM-Indexes is highly memory-bound, limiting performance.

PGM++ outperforms original PGM-Index and state-of-the-art learned indexes in speed, with up to 2.31x faster queries.

Abstract

Learned indexes have attracted significant research interest due to their ability to offer better space-time trade-offs compared to traditional B+-tree variants. Among various learned indexes, the PGM-Index based on error-bounded piecewise linear approximation is an elegant data structure that has demonstrated \emph{provably} superior performance over conventional B+-tree indexes. In this paper, we explore two interesting research questions regarding the PGM-Index: (a) \emph{Why are PGM-Indexes theoretically effective?} and (b) \emph{Why do PGM-Indexes underperform in practice?} For question~(a), we first prove that, for a set of $N$ sorted keys, the PGM-Index can, with high probability, achieve a lookup time of $O(\log\log N)$ while using $O(N)$ space. To the best of our knowledge, this is the \textbf{tightest bound} for learned indexes to date. For question~(b), we identify that querying PGM-Indexes is highly memory-bound, where the internal error-bounded search operations often become the bottleneck. To fill the performance gap, we propose PGM++, a \emph{simple yet effective} extension to the original PGM-Index that employs a mixture of different search strategies, with hyper-parameters automatically tuned through a calibrated cost model. Extensive experiments on real workloads demonstrate that PGM++ establishes a new Pareto frontier. At comparable space costs, PGM++ speeds up index lookup queries by up to $\mathbf{2.31\times}$ and $\mathbf{1.56\times}$ when compared to the original PGM-Index and state-of-the-art learned indexes.