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 $d$.

提出了一种基于混合学习算法的PAC学习方法，该算法可用于密度估计中的概率分布，其中包含了学习概率分布，学习混合分布等，其中混合分布包括轴向高斯混合分布，高斯混合分布和对数凹分布。

混合物的高效样本学习