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

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 $\varepsilon2-\varepsilon1$ 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.