Testing Juntas and Junta Subclasses with Relative Error

TL;DR

This work resolves relative-error junta testing by giving a near-optimal $\tilde{O}(k/\varepsilon)$-query tester for $k$-juntas and extends to any permutation-closed subclass ${\cal C}(k)$ with complexity $\tilde{O}\left( \frac{k\log|{\cal C}(k)^*|}{\varepsilon} \right)$. The approach combines implicit learning with an Approx$(h,\kappa)$ framework, partition-based identification of relevant variables, and two new subroutines FindVarValue and MapBack to align recovered variables under a hidden permutation. The results yield efficient relative-error testers for natural function classes such as size-$s$ decision trees, branching programs, and Boolean formulas, matching standard-model complexities up to polylog factors and advancing the understanding of sparse function testing in the relative-error setting. Overall, the paper bridges junta-testing techniques with relative-error distance to produce broadly applicable, scalable testers for sparse Boolean functions, with potential impact on theoretical and practical sparse-function detection.

Abstract

This papers considers the junta testing problem in a recently introduced ``relative error'' variant of the standard Boolean function property testing model. In relative-error testing we measure the distance from $f$ to $g$, where $f,g: \{0,1\}^n \to \{0,1\}$, by the ratio of $|f^{-1}(1) \triangle g^{-1}(1)|$ (the number of inputs on which $f$ and $g$ disagree) to $|f^{-1}(1)|$ (the number of satisfying assignments of $f$), and we give the testing algorithm both black-box access to $f$ and also access to independent uniform samples from $f^{-1}(1)$. Chen et al. (SODA 2025) observed that the class of $k$-juntas is $\text{poly}(2^k,1/ε)$-query testable in the relative-error model, and asked whether $\text{poly}(k,1/ε)$ queries is achievable. We answer this question affirmatively by giving a $\tilde{O}(k/ε)$-query algorithm, matching the optimal complexity achieved in the less challenging standard model. Moreover, as our main result, we show that any subclass of $k$-juntas that is closed under permuting variables is relative-error testable with a similar complexity. This gives highly efficient relative-error testing algorithms for a number of well-studied function classes, including size-$k$ decision trees, size-$k$ branching programs, and size-$k$ Boolean formulas.

Theorems & Definitions (59)

• Theorem 1: Relative-error junta testing
• Theorem 2: Relative-error testing of junta subclasses
• Corollary 3
• Remark 4: On distribution-free testing versus relative-error testing.
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8
• proof