Learning junta distributions is computationally equivalent to LPN, and a new algorithm achieves near-optimal sample complexity O((k/epsilon^2)(2^k + log n)).
Testing Junta Truncation
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abstract
We consider the basic statistical problem of detecting truncation of the uniform distribution on the Boolean hypercube by juntas. More concretely, we give upper and lower bounds on the problem of distinguishing between i.i.d. sample access to either (a) the uniform distribution over $\{0,1\}^n$, or (b) the uniform distribution over $\{0,1\}^n$ conditioned on the satisfying assignments of a $k$-junta $f: \{0,1\}^n\to\{0,1\}$. We show that (up to constant factors) $\min\{2^k + \log{n\choose k}, {2^{k/2}\log^{1/2}{n\choose k}}\}$ samples suffice for this task and also show that a $\log{n\choose k}$ dependence on sample complexity is unavoidable. Our results suggest that testing junta truncation requires learning the set of relevant variables of the junta.
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2025 1verdicts
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New Statistical and Computational Results for Learning Junta Distributions
Learning junta distributions is computationally equivalent to LPN, and a new algorithm achieves near-optimal sample complexity O((k/epsilon^2)(2^k + log n)).