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Querying in Constant Expected Time with Learned Indexes

Luis Croquevielle, Guang Yang, Liang Liang, Ali Hadian, Thomas Heinis

TL;DR

It is proved that 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.

Abstract

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 research, meaning our work generalizes previous results. 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.

Querying in Constant Expected Time with Learned Indexes

TL;DR

It is proved that 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.

Abstract

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 elements, it has been shown that learned indexes can find a key in expected time using at most linear space, compared with for non-learned methods. In this work, we prove 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 research, meaning our work generalizes previous results. 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.
Paper Structure (35 sections, 9 theorems, 23 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 35 sections, 9 theorems, 23 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Data
  4. Learning Problem
  5. Model of Computation
  6. Main Results and ESPC Index
  7. Complexity Bounds
  8. Equal-Split Piecewise Constant Index
  9. Complexity Proofs and Analysis
  10. Preliminary Results
  11. Main Probabilistic Bound
  12. Proof of Asymptotic Complexity
  13. Distribution of query parameter
  14. Unbounded support
  15. Analysis and Benchmarking
  16. ...and 20 more sections

Key Result

Theorem 1

Suppose $f$ has support $[a, b]$ and $\rho_{ f} < \infty$. Define $\rho=\log((b-a)\rho_{ f})$. Then, there is a procedure $R_n$ for building learned indexes such that $\overline{S}_n=O(n)$ and $\overline{T}_n=O(\rho)$. That is, for array $A$ with $n$ keys an index can be built with space overhead $O

Figures (5)

  • Figure 1: Predictive and corrective steps of a learned index.
  • Figure 2: Partition of key range into four equal-length intervals, with approximator of $\mathbf{rank}$.
  • Figure 3: Equal-length partitions for $[X_{(1)}, X_{(n)}]$ (given by $\{I_k\}$) and $[a, b]$ (given by $\{J_k\}$).
  • Figure 4: Average experimental error and theoretical bound for expected error, as functions of the number $K$ of subintervals. Top row shows synthetic datasets, bottom row shows real-world datasets. The plots use a logarithmic scale for both axes, which prevents clustering of the points.
  • Figure 5: $\hat{\rho}_f$ estimate for normal, amzn and osm datasets as number of samples $n$ increases. The estimation process uses the histogram method to approximate the density $f$.

Theorems & Definitions (9)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Lemma 4
  • Theorem 5
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Theorem 9