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 $m_{\mathcal{F}}(\varepsilon)$ in the realizable setting, we provide a method for learning $\mathcal F^k$ with sample complexity $O({k\log k \cdot m_{\mathcal F}(\varepsilon) }/{\varepsilon^{2}})$ in the agnostic setting. Our mixture learning algorithm has the property that, if the $\mathcal F$-learner is proper, then the $\mathcal F^k$-learner is proper as well. We provide two applications of our main result. 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 sample complexity $\widetilde{O}({kd}/{\epsilon ^ 4})$, which is tight in $k$ and $d$. 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 $\widetilde{O}({k^3d^2}/{\varepsilon ^ 4})$ and $\widetilde{O}(k^4d^4/\varepsilon ^ 2)$ in its dependence on $k$ and $d$.