Feature Selection and Junta Testing are Statistically Equivalent

For a function $f : \{0, 1\}^n \rightarrow \{0, 1\}$, 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 align ( 1 (2^k nk + nk ) ).align