# A o(d) · polylog~n Monotonicity Tester for Boolean Functions over the Hypergrid [n]d

arXiv: https://arxiv.org/abs/1710.10545 · HTML: https://mlsanity.com/paper/1710.10545/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2017.

## Built on by (most cited first, 2 of 2)

- [Agnostic proper learning of monotone functions: beyond the black-box correction barrier](https://mlsanity.com/paper/2304.02700/) (2023)
- [Uniformity Testing over Hypergrids with Subcube Conditioning](https://mlsanity.com/paper/2302.09013/) (2023)

## Abstract (arXiv)

We study monotonicity testing of Boolean functions over the hypergrid [n]d and design a non-adaptive tester with 1-sided error whose query complexity is tildeO(d5/6)· poly(log n,1/ε). Previous to our work, the best known testers had query complexity linear in d but independent of n. We improve upon these testers as long as n = 2^do(1). To obtain our results, we work with what we call the augmented hypergrid, which adds extra edges to the hypergrid. Our main technical contribution is a Margulis-style isoperimetric result for the augmented hypergrid, and our tester, like previous testers for the hypercube domain, performs directed random walks on this structure.