Sample-Efficient Learning of Mixtures

We consider PAC learning of probability distributions (a.k.a. density estimation), where we are given an i.i.d. sample generated from an unknown target distribution, and want to output a distribution that is close to the target in total variation distance. Let $\mathcal F$ be an arbitrary class of probability distributions, and let $\mathcal{F}^k$ denote the class of $k$-mixtures of elements of $\mathcal F$. Assuming the existence of a method for learning $\mathcal F$ with sample complexity $m_{\mathcal{F}}(\epsilon)$, we provide a method for learning $\mathcal F^k$ with sample complexity $O({k\log k \cdot m_{\mathcal F}(\epsilon) }/{\epsilon^{2}})$. Our mixture learning algorithm has the property that, if the $\mathcal F$-learner is proper/agnostic, then the $\mathcal F^k$-learner would be proper/agnostic as well. This general result enables us to improve the best known sample complexity upper bounds for a variety of important mixture classes. First, we show that the class of mixtures of $k$ axis-aligned Gaussians in $\mathbb{R}^d$ is PAC-learnable in the agnostic setting with $\widetilde{O}({kd}/{\epsilon ^ 4})$ samples, which is tight in $k$ and $d$ up to logarithmic factors. Second, we show that the class of mixtures of $k$ Gaussians in $\mathbb{R}^d$ is PAC-learnable in the agnostic setting with sample complexity $\widetilde{O}({kd^2}/{\epsilon ^ 4})$, which improves the previous known bounds of $\widetilde{O}({k^3d^2}/{\epsilon ^ 4})$ and $\widetilde{O}(k^4d^4/\epsilon ^ 2)$ in its dependence on $k$ and $d$. Finally, we show that the class of mixtures of $k$ log-concave distributions over $\mathbb{R}^d$ is PAC-learnable using $\widetilde{O}(d^{(d+5)/2}\epsilon^{-(d+9)/2}k)$ samples.Comment: A bug from the previous version, which appeared in AAAI 2018 proceedings, is fixed. 18 page

Hashing-Based-Estimators for Kernel Density in High Dimensions

Given a set of points $P\subset \mathbb{R}^{d}$ and a kernel $k$, the Kernel Density Estimate at a point $x\in\mathbb{R}^{d}$ is defined as $\mathrm{KDE}_{P}(x)=\frac{1}{|P|}\sum_{y\in P} k(x,y)$. We study the problem of designing a data structure that given a data set $P$ and a kernel function, returns *approximations to the kernel density* of a query point in *sublinear time*. We introduce a class of unbiased estimators for kernel density implemented through locality-sensitive hashing, and give general theorems bounding the variance of such estimators. These estimators give rise to efficient data structures for estimating the kernel density in high dimensions for a variety of commonly used kernels. Our work is the first to provide data-structures with theoretical guarantees that improve upon simple random sampling in high dimensions.Comment: A preliminary version of this paper appeared in FOCS 201