The problem of testing monotonicity for Boolean functions on the hypergrid, $f:[n]^d \to \{0,1\}$ is a classic topic in property testing. When $n=2$, the domain is the hypercube. For the hypercube case, a breakthrough result of Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making $\otilde(\eps^{-2}\sqrt{d})$ queries. Up to polylog $d$ and $\eps$ factors, this bound matches the $\widetilde{\Omega}(\sqrt{d})$-query non-adaptive lower bound (Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any $n > 2$, the optimal non-adaptive complexity was unknown. A previous result of the authors achieves a $\otilde(d^{5/6})$-query upper bound (SODA 2020), quite far from the $\sqrt{d}$ bound for the hypercube. In this paper, we resolve the non-adaptive complexity of monotonicity testing for all constant $n$, up to $\poly(\eps^{-1}\log d)$ factors. Specifically, we give a non-adaptive, one-sided monotonicity tester making $\otilde(\eps^{-2}n\sqrt{d})$ queries. From a technical standpoint, we prove new directed isoperimetric theorems over the hypergrid $[n]^d$. These results generalize the celebrated directed Talagrand inequalities that were only known for the hypercube.
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Testing k-Monotonicity
A Boolean {\em $k$-monotone} function defined over a finite poset domain ${\cal D}$
alternates between the values $0$ and $1$ at most $k$ times on any ascending chain in ${\cal D}$. Therefore, $k$-monotone functions are natural generalizations of the classical {\em monotone} functions, which are the {\em $1$-monotone} functions.
Motivated by the recent interest in $k$-monotone functions in the context of circuit complexity and learning theory, and by the central role that monotonicity testing plays in the context of property testing, we initiate a systematic study of $k$-monotone functions, in the property testing model. In this model, the goal is to distinguish functions that are $k$-monotone (or are close to being $k$-monotone) from functions that are far from being $k$-monotone.
Our results include the following:
\begin{enumerate}
\item We demonstrate a separation between testing $k$-monotonicity and testing monotonicity, on the hypercube domain $\{0,1\}^d$, for $k\geq 3$;
\item We demonstrate a separation between testing and learning on $\{0,1\}^d$, for $k=\omega(\log d)$: testing $k$-monotonicity can be performed with $2^{O(\sqrt d \cdot \log d\cdot \log{1/\eps})}$ queries, while learning $k$-monotone functions requires $2^{\Omega(k\cdot \sqrt d\cdot{1/\eps})}$ queries (Blais et al. (RANDOM 2015)).
\item We present a tolerant test for functions $f\colon[n]^d\to \{0,1\}$ with complexity independent of $n$, which makes progress on a problem left open by Berman et al. (STOC 2014).
\end{enumerate}
Our techniques exploit the testing-by-learning paradigm, use novel applications of Fourier analysis on the grid $[n]^d$, and draw connections to distribution testing techniques.
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- Award ID(s):
- 1649515
- NSF-PAR ID:
- 10033552
- Date Published:
- Journal Name:
- ITCS
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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