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Optimal Non-Adaptive Tolerant Junta Testing via Local Estimators

Shivam Nadimpalli, Shyamal Patel

TL;DR

The paper addresses the problem of tolerant, non-adaptive testing of k-juntas by designing a distance-estimator that distinguishes between functions ε1-close to a k-junta and ε2-far from any k-junta with high probability. Its key approach uses locally computable estimators of the mean absolute value |E[f]| based on small-radius Hamming balls, combined with flat polynomials and a hold-out noise operator to enable smoothing without adaptivity. The authors prove a tight upper bound of 2^{~O(√k log(1/(ε2−ε1)))} queries for the non-adaptive tolerant tester and complement it with a matching lower bound, establishing the first tight results for a natural tolerant Boolean-function property. The work also introduces a broadly applicable local-estimation framework, with potential implications for approximation theory and pseudorandomness in boolean function analysis.

Abstract

We give a non-adaptive algorithm that makes $2^{\tilde{O}(\sqrt{k\log(1/\varepsilon_2 - \varepsilon_1)})}$ queries to a Boolean function $f:\{\pm 1\}^n \rightarrow \{\pm 1\}$ and distinguishes between $f$ being $\varepsilon_1$-close to some $k$-junta versus $\varepsilon_2$-far from every $k$-junta. At the heart of our algorithm is a local mean estimation procedure for Boolean functions that may be of independent interest. We complement our upper bound with a matching lower bound, improving a recent lower bound obtained by Chen et al. We thus obtain the first tight bounds for a natural property of Boolean functions in the tolerant testing model.

Optimal Non-Adaptive Tolerant Junta Testing via Local Estimators

TL;DR

The paper addresses the problem of tolerant, non-adaptive testing of k-juntas by designing a distance-estimator that distinguishes between functions ε1-close to a k-junta and ε2-far from any k-junta with high probability. Its key approach uses locally computable estimators of the mean absolute value |E[f]| based on small-radius Hamming balls, combined with flat polynomials and a hold-out noise operator to enable smoothing without adaptivity. The authors prove a tight upper bound of 2^{~O(√k log(1/(ε2−ε1)))} queries for the non-adaptive tolerant tester and complement it with a matching lower bound, establishing the first tight results for a natural tolerant Boolean-function property. The work also introduces a broadly applicable local-estimation framework, with potential implications for approximation theory and pseudorandomness in boolean function analysis.

Abstract

We give a non-adaptive algorithm that makes queries to a Boolean function and distinguishes between being -close to some -junta versus -far from every -junta. At the heart of our algorithm is a local mean estimation procedure for Boolean functions that may be of independent interest. We complement our upper bound with a matching lower bound, improving a recent lower bound obtained by Chen et al. We thus obtain the first tight bounds for a natural property of Boolean functions in the tolerant testing model.
Paper Structure (27 sections, 22 theorems, 100 equations, 2 figures, 1 table, 4 algorithms)

This paper contains 27 sections, 22 theorems, 100 equations, 2 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Junta Testing.
  3. Tolerant Junta Testing.
  4. Technical Overview
  5. Tolerant Junta Testing via Local Estimators.
  6. A Matching Lower Bound.
  7. Discussion
  8. Preliminaries
  9. Boolean Functions
  10. Fourier Analysis over $\{\pm1 \}^n$
  11. Coordinate Oracles
  12. Flat Polynomials
  13. Numerical Differentiation
  14. Local Estimators
  15. Locally Estimating $|\operatorname{{\bf E}}[f]|$
  16. ...and 12 more sections

Key Result

Theorem 2

There exists a non-adaptive $\varepsilon$-distance estimator for the set of $k$-juntas that makes at most $\mathrm{poly}(k, \varepsilon^{-1}) \cdot 2^{\widetilde{O}(\sqrt{k \log(1/\varepsilon)})}$ queries, where the $\widetilde{O}$ notation hides $\log(k)$ and $\log\log(1/\varepsilon)$ factors.

Figures (2)

  • Figure 1: A draw of $\boldsymbol{f}_{\mathrm{yes}}\sim\mathcal{D}_{{\mathrm{yes}}}$. The left hand side depicts the control subcube $\{\pm1 \}^{{\boldsymbol{C}}}$, the right hand side depicts an action subcube $\{\pm1 \}^{{\boldsymbol{A}}}$ and the dashed lines indicate Hamming weight levels. The functions ${\boldsymbol{b}}_1,{\boldsymbol{b}}_2:\{\pm1 \}^{{\boldsymbol{C}}}\to \{\pm1 \}$ are random functions on the control subcube.
  • Figure 2: A draw of $\boldsymbol{f}_{\mathrm{no}}\sim\mathcal{D}_{{\mathrm{no}}}$. Our conventions are as in \ref{['fig:yes']}.

Theorems & Definitions (44)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Definition 7
  • Proposition 9
  • Lemma 10: Theorem 2.1 of kahn1996inclusion
  • Lemma 11
  • Lemma 12
  • Theorem 13
  • ...and 34 more