Feature Selection and Junta Testing are Statistically Equivalent

For a function $f \colon \{0,1\}^n \to \{0,1\}$, the junta testing problem asks whether $f$ depends on only $k$ variables. If $f$ depends on only $k$ variables, the feature selection problem asks to find those variables. We prove that these two tasks are statistically equivalent. Specifically, we show that the ``brute-force'' algorithm, which checks for any set of $k$ variables consistent with the sample, is simultaneously sample-optimal for both problems, and the optimal sample size is $$\Theta\left(\frac 1 \varepsilon \left( \sqrt{2^k \log {n \choose k}} + \log {n \choose k}\right)\right).$$