On the Structure, Covering, and Learning of Poisson Multinomial Distributions

An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We prove a structural characterization of these distributions, showing that, for all $\varepsilon >0$, any $(n, k)$-Poisson multinomial random vector is $\varepsilon$-close, in total variation distance, to the sum of a discretized multidimensional Gaussian and an independent $(\text{poly}(k/\varepsilon), k)$-Poisson multinomial random vector. Our structural characterization extends the multi-dimensional CLT of Valiant and Valiant, by simultaneously applying to all approximation requirements $\varepsilon$. In particular, it overcomes factors depending on $\log n$ and, importantly, the minimum eigenvalue of the PMD's covariance matrix from the distance to a multidimensional Gaussian random variable. We use our structural characterization to obtain an $\varepsilon$-cover, in total variation distance, of the set of all $(n, k)$-PMDs, significantly improving the cover size of Daskalakis and Papadimitriou, and obtaining the same qualitative dependence of the cover size on $n$ and $\varepsilon$ as the $k=2$ cover of Daskalakis and Papadimitriou. We further exploit this structure to show that $(n,k)$-PMDs can be learned to within $\varepsilon$ in total variation distance from $\tilde{O}_k(1/\varepsilon^2)$ samples, which is near-optimal in terms of dependence on $\varepsilon$ and independent of $n$. In particular, our result generalizes the single-dimensional result of Daskalakis, Diakonikolas, and Servedio for Poisson Binomials to arbitrary dimension.Comment: 49 pages, extended abstract appeared in FOCS 201

Sampling Correctors

In many situations, sample data is obtained from a noisy or imperfect source. In order to address such corruptions, this paper introduces the concept of a sampling corrector. Such algorithms use structure that the distribution is purported to have, in order to allow one to make "on-the-fly" corrections to samples drawn from probability distributions. These algorithms then act as filters between the noisy data and the end user. We show connections between sampling correctors, distribution learning algorithms, and distribution property testing algorithms. We show that these connections can be utilized to expand the applicability of known distribution learning and property testing algorithms as well as to achieve improved algorithms for those tasks. As a first step, we show how to design sampling correctors using proper learning algorithms. We then focus on the question of whether algorithms for sampling correctors can be more efficient in terms of sample complexity than learning algorithms for the analogous families of distributions. When correcting monotonicity, we show that this is indeed the case when also granted query access to the cumulative distribution function. We also obtain sampling correctors for monotonicity without this stronger type of access, provided that the distribution be originally very close to monotone (namely, at a distance $O(1/\log^2 n)$). In addition to that, we consider a restricted error model that aims at capturing "missing data" corruptions. In this model, we show that distributions that are close to monotone have sampling correctors that are significantly more efficient than achievable by the learning approach. We also consider the question of whether an additional source of independent random bits is required by sampling correctors to implement the correction process