Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism

Abstract

The function f : {−1, 1} n → {−1, 1} is a k-junta if it depends on at most k of its variables. We consider the problem of tolerant testing of k-juntas, where the testing algorithm must accept any function that is -close to some k-junta and reject any function that is -far from every k -junta for some = O( ) and k = O(k).Our first result is an algorithm that solves this problem with query complexity polynomial in k and 1/ . This result is obtained via a new polynomialtime approximation algorithm for submodular function minimization (SFM) under large cardinality constraints, which holds even when only given an approximate oracle access to the function.Our second result considers the case where k = k. We show how to obtain a smooth tradeoff between the amount of tolerance and the query complexity in this setting. Specifically, we design an algorithm that given ρ ∈ (0, 1/2) accepts any function that is ρ 8 -close to some k-junta and rejects any function that is -far from every k-junta. The query complexity of the algorithm is OFinally, we show how to apply the second result to the problem of tolerant isomorphism testing between two unknown Boolean functions f and g. We give an algorithm for this problem whose query complexity only depends on the (unknown) smallest k such that either f or g is close to being a k-junta.

Junta Correlation is Testable

“…Input: x ∈ {±1} n , and an oracle D for S Output: an influence-testing sample at x with respect to S 1 Define (for z ∈ {±1} n ) I(z) = {g ∈ D : g(z) = g(x)} ; 2 Initialize X = ∅ ; 3 while |X | < |D| do 4 Let y be a copy of x, but flip each bit independently with probability 1 2|D| Proof. Each element that Influence-testing-sample adds to X differs from x on exactly one coordinate of S. Moreover, the test on line 5 ensures that every coordinate of S is represented by at most one element of X .…”

“…In the opposite (i.e., algorithmic) direction, Blais et al [4] proved the following theorem. Theorem 1.2.…”

“…g : g ∈ D} be an influence testing sample at x (i) (from Influence-testing-sample), where y (i) g is the element for which g(y (i) ) = g(x (i) ) ; 4 For g ∈ D, let Inf g = 1 2 ; 5 return D ′ = {g ∈ D : Inf g ≥ 3 2 t} Algorithm 5: Threshold-influences Lemma 5.8. Assume that D is an oracle for S and that f is a S-junta.…”

Approximating the distance to monotonicity of Boolean functions

“…Beyond the (standard) property testing, the questions of erasure‐resilient and tolerant testing have also received some attention ([32, 49, 57] study the erasure‐resilient model, and [2, 6, 7, 14, 18, 31, 35, 37, 40, 43, 47, 50, 51, 55, 63] study the tolerant testing model). Specifically for monotonicity, in [32], an erasure‐resilient tester for functions on hypergrids is designed.…”

“…Using the connection between distance approximation and erasure‐resilient testing, our approximation algorithm implies an erasure‐resilient tester that has a less stringent restriction on , specifically, . For approximating the distance to ‐juntas [14, 31], the best algorithm with additive error of makes queries [31], and the best lower bound was for nonadaptive algorithms [50].…”