Tight Bounds for the Distribution-Free Testing of Monotone Conjunctions

We improve both upper and lower bounds for the distribution-free testing of monotone conjunctions. Given oracle access to an unknown Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ and sampling oracle access to an unknown distribution $\mathcal{D}$ over $\{0,1\}^n$, we present an $\tilde{O}(n^{1/3}/\epsilon^5)$-query algorithm that tests whether $f$ is a monotone conjunction versus $\epsilon$-far from any monotone conjunction with respect to $\mathcal{D}$. This improves the previous best upper bound of $\tilde{O}(n^{1/2}/\epsilon)$ by Dolev and Ron when $1/\epsilon$ is small compared to $n$. For some constant $\epsilon_0>0$, we also prove a lower bound of $\tilde{\Omega}(n^{1/3})$ for the query complexity, improving the previous best lower bound of $\tilde{\Omega}(n^{1/5})$ by Glasner and Servedio. Our upper and lower bounds are tight, up to a poly-logarithmic factor, when the distance parameter $\epsilon$ is a constant. Furthermore, the same upper and lower bounds can be extended to the distribution-free testing of general conjunctions, and the lower bound can be extended to that of decision lists and linear threshold functions.