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Efficient Testing Implies Structured Symmetry

Cynthia Dwork, Pranay Tankala

TL;DR

An equivalence is shown between properties that are efficiently testable from few samples and properties with structured symmetry, which depend only on the function's average values on parts of a low-complexity partition of the domain.

Abstract

Given a small random sample of $n$-bit strings labeled by an unknown Boolean function, which properties of this function can be tested computationally efficiently? We show an equivalence between properties that are efficiently testable from few samples and properties with structured symmetry, which depend only on the function's average values on parts of a low-complexity partition of the domain. Without the efficiency constraint, a similar characterization in terms of unstructured symmetry was obtained by Blais and Yoshida (2019). Our main technical tool is supersimulation, which builds on methods from the algorithmic fairness literature to approximate arbitrarily complex functions by small-circuit simulators that fool significantly larger distinguishers. We extend the characterization along other axes as well. We show that allowing parts to overlap exponentially reduces their required number, broadening the scope of the construction from properties testable with $O(\log n)$ samples to properties testable with $O(n)$ samples. For larger sample sizes, we show that any efficient tester is essentially checking for indistinguishability from a bounded collection of small circuits, in the spirit of a characterization of testable graph properties. Finally, we show that our results for Boolean function testing generalize to high-entropy distribution testing on arbitrary domains.

Efficient Testing Implies Structured Symmetry

TL;DR

An equivalence is shown between properties that are efficiently testable from few samples and properties with structured symmetry, which depend only on the function's average values on parts of a low-complexity partition of the domain.

Abstract

Given a small random sample of -bit strings labeled by an unknown Boolean function, which properties of this function can be tested computationally efficiently? We show an equivalence between properties that are efficiently testable from few samples and properties with structured symmetry, which depend only on the function's average values on parts of a low-complexity partition of the domain. Without the efficiency constraint, a similar characterization in terms of unstructured symmetry was obtained by Blais and Yoshida (2019). Our main technical tool is supersimulation, which builds on methods from the algorithmic fairness literature to approximate arbitrarily complex functions by small-circuit simulators that fool significantly larger distinguishers. We extend the characterization along other axes as well. We show that allowing parts to overlap exponentially reduces their required number, broadening the scope of the construction from properties testable with samples to properties testable with samples. For larger sample sizes, we show that any efficient tester is essentially checking for indistinguishability from a bounded collection of small circuits, in the spirit of a characterization of testable graph properties. Finally, we show that our results for Boolean function testing generalize to high-entropy distribution testing on arbitrary domains.

Paper Structure

This paper contains 30 sections, 18 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Results Overview
  4. Extensions
  5. Technical Overview
  6. Step 1: Simulating the Oracle
  7. Step 2: Simulating the Tester
  8. Step 3: Supersimulators
  9. Related Work
  10. Structure vs Randomness
  11. Algorithmic Fairness
  12. Property Testing
  13. Preliminaries
  14. Property Testing
  15. Structured Symmetry
  16. ...and 15 more sections

Key Result

Theorem 1.1

[theorem]thm:blais If a property $\mathcal{P}$ of Boolean functions is testable with proximity $\varepsilon$ using $m$ samples, then $\mathcal{P} \subseteq \mathcal{Q} \subseteq \mathcal{P}_{\varepsilon}$ for some $2^{2^{O(m)}}$-part symmetric property $\mathcal{Q}$.Note that $m$ may scale with $n$

Figures (1)

  • Figure 1: Illustration of the supersimulator construction of \ref{['sec:symmetry-thm']}. Plot (\ref{['fig:tester']}) depicts a deterministic $3$-sample tester $T$ by the set of triples that cause it to output Accept. A one-way restriction of $T$ is shown in blue. Plot (\ref{['fig:before']}) depicts a simulator in the sequence, with translucent regions indicating fractional estimates for $T$'s Accept region. Plot (\ref{['fig:after']}) depicts the next simulator in the sequence, after an update has been performed based on the chosen restriction of $T$.

Theorems & Definitions (38)

  • Theorem 1.1: blais2019testable
  • Theorem 1.1
  • Theorem 1.1
  • Definition 2.1: Sample-Testable Property
  • Definition 2.2: Regularity and Indistinguishability
  • Definition 2.3: Structured Sums
  • Lemma 2.4: Complexity-Theoretic Regularity trevisan2009regularity
  • Definition 2.5
  • Lemma 2.5: Supersimulators dwork2025supersimulators
  • Theorem 3.0
  • ...and 28 more