Junta Distance Approximation with Sub-Exponential Queries

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the \emph{tolerant testing} of juntas. Given black-box access to a Boolean function $f:{\pm1}^{n} \to {\pm1}$, we give a $poly(k, \frac{1}{\varepsilon})$ query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from $k'$-juntas, where $k' = O(\frac{k}{\varepsilon^2})$. In the non-relaxed setting, we extend our ideas to give a $2^{{\tilde{O}(\sqrt{k/\varepsilon})}$} (adaptive) query algorithm that distinguishes between functions that are $\gamma$-close to $k$-juntas and $(\gamma+\varepsilon)$-far from $k$-juntas. To the best of our knowledge, this is the first subexponential-in-$k$ query algorithm for approximating the distance of $f$ to being a $k$-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in $k$). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [AoP, 1994].