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Mildly Exponential Lower Bounds on Tolerant Testers for Monotonicity, Unateness, and Juntas
Authors
Xi Chen
Anindya De
+3Â more
Yuhao Li
Shivam Nadimpalli
Rocco A. Servedio
Publication date
21 September 2023
Publisher
View
on
arXiv
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
Ω
(
n
1
/
4
/
ε
)
2^{\Omega(n^{1/4}/\sqrt{\varepsilon})}
2
Ω
(
n
1/4
/
ε
​
)
-query lower bound for non-adaptive, two-sided tolerant monotonicity testers and unateness testers when the "gap" parameter
ε
2
−
ε
1
\varepsilon_2-\varepsilon_1
ε
2
​
−
ε
1
​
is equal to
ε
\varepsilon
ε
, for any
ε
≥
1
/
n
\varepsilon \geq 1/\sqrt{n}
ε
≥
1/
n
​
;
∙
\bullet
∙
A
2
Ω
(
k
1
/
2
)
2^{\Omega(k^{1/2})}
2
Ω
(
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
Ω
~
(
n
3
/
2
)
\tilde{\Omega}(n^{3/2})
Ω
~
(
n
3/2
)
queries, and the best prior lower bound known for juntas was
p
o
l
y
(
k
)
\mathrm{poly}(k)
poly
(
k
)
queries.Comment: 20 pages, 1 figur
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oai:arXiv.org:2309.12513
Last time updated on 12/10/2023