# Almost Optimal Distribution-free Junta Testing

## TL;DR

This paper introduces a simpler, more efficient adaptive algorithm for distribution-free $k$-junta testing that reduces query complexity from roughly $k^2$ to $k$, improving the practicality of property testing.

## Contribution

It presents a new two-sided error adaptive algorithm for distribution-free $k$-junta testing with nearly optimal query complexity of $ ilde O(k/\epsilon)$, improving upon previous methods.

## Key findings

- Reduces query complexity from $ ilde O(k^2/\epsilon)$ to $ ilde O(k/\epsilon)$.
- Provides a simpler and more efficient testing algorithm.
- Achieves near-optimal performance in distribution-free junta testing.

## Abstract

We consider the problem of testing whether an unknown $n$-variable Boolean function is a $k$-junta in the distribution-free property testing model, where the distance between function is measured with respect to an arbitrary and unknown probability distribution over $\{0,1\}^n$. Chen, Liu, Servedio, Sheng and Xie showed that the distribution-free $k$-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes $\tilde O(k^2)/\epsilon$ queries. In this paper, we give a simple two-sided error adaptive algorithm that makes $\tilde O(k/\epsilon)$ queries.