Mildly Exponential Lower Bounds on Tolerant Testers for Monotonicity, Unateness, and Juntas

TL;DR

This paper establishes the first super-polynomial lower bounds for tolerant property testing of monotonicity, unateness, and juntas, demonstrating inherent complexity in these testing problems even with constant gaps.

Contribution

It provides the first mildly exponential lower bounds for tolerant testing of monotonicity, unateness, and juntas, advancing understanding of their computational hardness.

Findings

Super-polynomial lower bounds for tolerant monotonicity testing

Super-polynomial lower bounds for tolerant unateness testing

Super-polynomial lower bounds for tolerant juntas testing

Abstract

We give the first super-polynomial (in fact, mildly exponential) lower bounds for tolerant testing (equivalently, distance estimation) of monotonicity, unateness, and juntas with a constant separation between the "yes" and "no" cases. Specifically, we give $\bullet$ A $2^{\Omega(n^{1/4}/\sqrt{\varepsilon})}$-query lower bound for non-adaptive, two-sided tolerant monotonicity testers and unateness testers when the "gap" parameter $\varepsilon_2-\varepsilon_1$ is equal to $\varepsilon$, for any $\varepsilon \geq 1/\sqrt{n}$; $\bullet$ A $2^{\Omega(k^{1/2})}$-query lower bound for non-adaptive, two-sided tolerant junta testers when the gap parameter is an absolute constant. In the constant-gap regime no non-trivial prior lower bound was known for monotonicity, the best prior lower bound known for unateness was $\tilde{\Omega}(n^{3/2})$ queries, and the best prior lower bound known for juntas was $\mathrm{poly}(k)$ queries.