学习泊松二项分布
Learning Poisson Binomial Distributions
摘要：
我们考虑无监督学习中的一个基本问题：学习未知的\emph{Poisson Binomial Distribution}。$\{0,1,\dots,n\}$上的泊松二项分布（Poisson Binomial Distribution，PBD）是$n$独立伯努利随机变量之和的分布，这些随机变量可能具有任意的、可能不相等的期望。这些分布首先由S. Poisson在1837年\cite{Poisson:37}和是一个自然的$n$参数推广熟悉的二项分布。令人惊讶的是，在我们开展工作之前，人们对这个基本学习问题知之甚少，而且已知的结果也远非最佳。我们基本上解决了这类基本分布的学习问题的复杂性。作为我们的第一个主要结果，我们给出了一个高效的算法，学习$\eps$的准确性（相对于总的变化距离）使用$\tilde{O}(1/\eps^3)$样本\emph{independent of $n$}。该算法的运行时间是其输入数据大小的\emph{quasilinear}，即，$\tilde{O}(\log(n)/\eps^3)$位操作。（注意，从分布中提取的每个数据都是$\log(n)$位字符串。）我们的第二个主要结果是{ \em proper}学习算法，它使用$\tilde{O}(1/\eps^2)$样本学习$\eps$ -准确度，并在时间$(1/\eps)^{\poly (\log (1/\eps))} \cdot \log n$中运行。这几乎是最优的，因为任何算法{对于这个问题}必须使用$Ω(1/\eps^2)$样本。我们还给出了积极和消极的结果，这个学习问题的一些扩展独立的Bernoulli随机变量的加权和。
英文摘要：
We consider a basic problem in unsupervised learning: learning an unknown \emph{Poisson Binomial Distribution}. A Poisson Binomial Distribution (PBD) over $\{0,1,\dots,n\}$ is the distribution of a sum of $n$ independent Bernoulli random variables which may have arbitrary, potentially non-equal, expectations. These distributions were first studied by S. Poisson in 1837 \cite{Poisson:37} and are a natural $n$-parameter generalization of the familiar Binomial Distribution. Surprisingly, prior to our work this basic learning problem was poorly understood, and known results for it were far from optimal. We essentially settle the complexity of the learning problem for this basic class of distributions. As our first main result we give a highly efficient algorithm which learns to $\eps$-accuracy (with respect to the total variation distance) using $\tilde{O}(1/\eps^3)$ samples \emph{independent of $n$}. The running time of the algorithm is \emph{quasilinear} in the size of its input data, i.e., $\tilde{O}(\log(n)/\eps^3)$ bit-operations. (Observe that each draw from the distribution is a $\log(n)$-bit string.) Our second main result is a {\em proper} learning algorithm that learns to $\eps$-accuracy using $\tilde{O}(1/\eps^2)$ samples, and runs in time $(1/\eps)^{\poly (\log (1/\eps))} \cdot \log n$. This is nearly optimal, since any algorithm {for this problem} must use $Ω(1/\eps^2)$ samples. We also give positive and negative results for some extensions of this learning problem to weighted sums of independent Bernoulli random variables.
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