Near-Optimal Degree Testing for Bayes Nets
This work addresses a fundamental testing problem in Bayesian network analysis, providing near-optimal sample bounds for researchers in machine learning and statistics, though it is incremental in building on existing testing-by-learning frameworks.
The paper tackles the problem of determining the maximum in-degree of a Bayes net from an unknown distribution using sample access, establishing that the sample complexity is approximately Θ(2^{n/2}/ε^2). It achieves this by developing new algorithms for near-proper learning and high-probability learning under χ^2 divergence.
This paper considers the problem of testing the maximum in-degree of the Bayes net underlying an unknown probability distribution $P$ over $\{0,1\}^n$, given sample access to $P$. We show that the sample complexity of the problem is $\tildeΘ(2^{n/2}/\varepsilon^2)$. Our algorithm relies on a testing-by-learning framework, previously used to obtain sample-optimal testers; in order to apply this framework, we develop new algorithms for ``near-proper'' learning of Bayes nets, and high-probability learning under $χ^2$ divergence, which are of independent interest.