# Adaptivity is exponentially powerful for testing monotonicity of halfspaces

## TL;DR

This paper presents an adaptive testing algorithm for monotonicity of halfspaces that significantly reduces query complexity compared to non-adaptive methods, demonstrating exponential improvements.

## Contribution

It introduces a polynomial-logarithmic query adaptive algorithm for monotonicity testing of halfspaces, surpassing previous non-adaptive approaches.

## Key findings

- Adaptive algorithms require exponentially fewer queries than non-adaptive ones.
- The new algorithm operates in poly(log n, 1/ε) query complexity.
- Non-adaptive algorithms need almost Ω(n^{1/2}) queries for the same task.

## Abstract

We give a $\mathrm{poly}(\log n, 1/\epsilon)$-query adaptive algorithm for testing whether an unknown Boolean function $f: \{-1,1\}^n \to \{-1,1\}$, which is promised to be a halfspace, is monotone versus $\epsilon$-far from monotone. Since non-adaptive algorithms are known to require almost $\Omega(n^{1/2})$ queries to test whether an unknown halfspace is monotone versus far from monotone, this shows that adaptivity enables an exponential improvement in the query complexity of monotonicity testing for halfspaces.