Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps

TL;DR

This paper introduces optimal adaptive and nonadaptive algorithms for testing unateness of real-valued functions over hypercubes and grids, demonstrating that adaptivity provides a significant advantage in this setting.

Contribution

It provides the first optimal unateness testers for real-valued functions on hypercubes and grids, establishing the benefit of adaptivity over nonadaptivity.

Findings

Adaptive tester has query complexity $O(d/\epsilon)$.

Nonadaptive tester has query complexity $O((d/\epsilon) imes ext{log}(d/\epsilon))$.

Both testers are proven optimal for fixed $\epsilon$.

Abstract

We study the problem of testing unateness of functions $f:\{0,1\}^d \to \mathbb{R}.$ We give a $O(\frac{d}{\epsilon} \cdot \log\frac{d}{\epsilon})$-query nonadaptive tester and a $O(\frac{d}{\epsilon})$-query adaptive tester and show that both testers are optimal for a fixed distance parameter $\epsilon$. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. We also generalize our results to obtain optimal unateness testers for functions $f:[n]^d \to \mathbb{R}$. Our results establish that adaptivity helps with testing unateness of real-valued functions on domains of the form $\{0,1\}^d$ and, more generally, $[n]^d$. This stands in contrast to the situation for monotonicity testing where there is no adaptivity gap for functions $f:[n]^d \to \mathbb{R}$.