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- W2621731111 abstract "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 $O({klog k cdot m_{mathcal F}(epsilon) }/{epsilon^{2}})$. 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." @default.
- W2621731111 created "2017-06-15" @default.
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- W2621731111 date "2017-06-06" @default.
- W2621731111 modified "2023-09-27" @default.
- W2621731111 title "Sample-Efficient Learning of Mixtures" @default.
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