Despite its popularity in practice, theoretical guarantees for high-dimensional problems are only available under strong structural assumptions (e.g., sparsity).

We study the following independence testing problem: given access to samples from a distribution $P$ over $\{0,1\}^n$, decide whether $P$ is a product distribution or whether it is $\varepsilon$-far in total variation distance from any product distribution. For arbitrary distributions, this problem requires $\exp(n)$ samples. We show in this work that if $P$ has a sparse structure, then in fact only linearly many samples are required.Specifically, if $P$ is Markov with respect to a Bayesian network whose underlying DAG has in-degree bounded by $d$, then $\tilde{\Theta}(2^{d/2}\cdot n/\varepsilon^2)$ samples are necessary and sufficient for independence testing.

In this work we present novel differentially private identity (goodness-of-fit) testers for natural and widely studied classes of multivariate product distributions: Gaussians in R^d with known covariance and product distributions over {\pm 1}^d. Our testers have improved sample complexity compared to those derived from previous techniques, and are the first testers whose sample complexity matches the order-optimal minimax sample complexity of O(d^1/2/alpha^2) in many parameter regimes. We construct two types of testers, exhibiting tradeoffs between sample complexity and computational complexity. Finally, we provide a two-way reduction between testing a subclass of multivariate product distributions and testing univariate distributions, and thereby obtain upper and lower bounds for testing this subclass of product distributions.

The convention used will be clear depending on the context. Appendix E. A.1 Privacy Preliminaries We define differential privacy and state its closure under post-processing property. When δ = 0, we say that M satisfies ε-differential privacy or pure differential privacy. The DP Assouad's method provides a lower bound for the minimax risk by considering the problem In this work, we make frequent use of exponential families due to their expressive power. We appeal to the following properties of exponential families.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Planning in partially observable Markov decision processes is a central problem in AI which is known to be computationally hard.