Efficient Density Estimation via Piecewise Polynomial Approximation

We give a highly efficient "semi-agnostic" algorithm for learning univariate probability distributions that are well approximated by piecewise polynomial density functions. Let $p$ be an arbitrary distribution over an interval $I$ which is $\tau$-close (in total variation distance) to an unknown probability distribution $q$ that is defined by an unknown partition of $I$ into $t$ intervals and $t$ unknown degree-$d$ polynomials specifying $q$ over each of the intervals. We give an algorithm that draws \tilde{O}(t\new{(d+1)}/\eps^2) samples from $p$, runs in time \poly(t,d,1/\eps), and with high probability outputs a piecewise polynomial hypothesis distribution $h$ that is (O(\tau)+\eps)-close (in total variation distance) to $p$. This sample complexity is essentially optimal; we show that even for $\tau=0$, any algorithm that learns an unknown $t$-piecewise degree-$d$ probability distribution over $I$ to accuracy \eps must use \Omega({\frac {t(d+1)} {\poly(1 + \log(d+1))}} \cdot {\frac 1 {\eps^2}}) samples from the distribution, regardless of its running time. Our algorithm combines tools from approximation theory, uniform convergence, linear programming, and dynamic programming. We apply this general algorithm to obtain a wide range of results for many natural problems in density estimation over both continuous and discrete domains. These include state-of-the-art results for learning mixtures of log-concave distributions; mixtures of $t$-modal distributions; mixtures of Monotone Hazard Rate distributions; mixtures of Poisson Binomial Distributions; mixtures of Gaussians; and mixtures of $k$-monotone densities. Our general technique yields computationally efficient algorithms for all these problems, in many cases with provably optimal sample complexities (up to logarithmic factors) in all parameters

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