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Distribution-Free Testing of Decision Lists with a Sublinear Number of Queries

Xi Chen, Yumou Fei, Shyamal Patel

TL;DR

This work resolves key questions in distribution-free testing by establishing the first sublinear tester for decision lists, with query complexity $\tilde{O}(n^{11/12}/\varepsilon^3)$ and matching runtime. The authors develop a sketch-based framework that distinguishes monotone decision lists from general ones by analyzing a constructed auxiliary structure and classifying cycles into long and local types, and then reduce general decision-list testing to monotone-case testing. A sequence of birthday-paradox lemmas for bipartite graphs and hypergraphs underpins the sampling guarantees, with reductions that yield strong lower bounds: $\tilde{\Omega}(\sqrt{n})$ queries and $\tilde{\Omega}(n)$ samples in respective settings. Collectively, the results illuminate a fundamental separation between testing and learning for decision lists under distribution-free constraints and provide algorithmic tools for sublinear property testing in combinatorial Boolean classes.

Abstract

We give a distribution-free testing algorithm for decision lists with $\tilde{O}(n^{11/12}/\varepsilon^3)$ queries. This is the first sublinear algorithm for this problem, which shows that, unlike halfspaces, testing is strictly easier than learning for decision lists. Complementing the algorithm, we show that any distribution-free tester for decision lists must make $\tildeΩ(\sqrt{n})$ queries, or draw $\tildeΩ(n)$ samples when the algorithm is sample-based.

Distribution-Free Testing of Decision Lists with a Sublinear Number of Queries

TL;DR

This work resolves key questions in distribution-free testing by establishing the first sublinear tester for decision lists, with query complexity and matching runtime. The authors develop a sketch-based framework that distinguishes monotone decision lists from general ones by analyzing a constructed auxiliary structure and classifying cycles into long and local types, and then reduce general decision-list testing to monotone-case testing. A sequence of birthday-paradox lemmas for bipartite graphs and hypergraphs underpins the sampling guarantees, with reductions that yield strong lower bounds: queries and samples in respective settings. Collectively, the results illuminate a fundamental separation between testing and learning for decision lists under distribution-free constraints and provide algorithmic tools for sublinear property testing in combinatorial Boolean classes.

Abstract

We give a distribution-free testing algorithm for decision lists with queries. This is the first sublinear algorithm for this problem, which shows that, unlike halfspaces, testing is strictly easier than learning for decision lists. Complementing the algorithm, we show that any distribution-free tester for decision lists must make queries, or draw samples when the algorithm is sample-based.
Paper Structure (20 sections, 48 theorems, 114 equations, 2 figures, 22 algorithms)

This paper contains 20 sections, 48 theorems, 114 equations, 2 figures, 22 algorithms.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Preliminaries
  4. Birthday Paradox Lemmas
  5. Warm-up: Testing Total Orderings
  6. Sketches
  7. The order graph and classification of cycles
  8. Putting it all together: An $\tilde{O}(\sqrt{n}/\varepsilon)$ tester for total orderings
  9. Testing Algorithm for Monotone Decision Lists
  10. Preprocessing
  11. $\textsc{MaxIndex}$
  12. The auxiliary graph and classification of its cycles
  13. The main testing algorithm and proof of Theorem \ref{['theo:monotonealg']}
  14. Main procedures and their analyses
  15. Testing Algorithm for Decision Lists
  16. ...and 5 more sections

Key Result

Theorem 1.1

There is a two-sided, adaptive, distribution-free testing algorithm for decision lists that makes $\tilde{O}(n^{11/12}/\varepsilon^3)$ queries and has the same running time.For the running time we assume that standard bitwise operations such as bitwise AND, OR and XOR over $n$-bit strings each cost

Figures (2)

  • Figure 1: One-side Lower Bound Construction for Total Orderings. An edge from $x$ to $y$ indicates that $x <_\sigma y$. The solid edges in the figure denote those in the support of $\mathcal{D}_{NO}$.
  • Figure : $\textsc{Sketch} (f,T)$

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.0
  • Lemma 2.0
  • Theorem 3.1
  • Definition 3.2: Sketch
  • Lemma 3.3
  • proof
  • ...and 91 more