  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.
