We investigate the problem of identity testing for multidimensional histogram distributions. A distribution @math , where @math , is called a @math -histogram if there exists a partition of the domain into @math axis-aligned rectangles such that @math is constant within each such rectangle. Histograms are one of the most fundamental non-parametric families of distributions and have been extensively studied in computer science and statistics. We give the first identity tester for this problem with sub-learning sample complexity in any fixed dimension and a nearly-matching sample complexity lower bound. More specifically, let @math be an unknown @math -dimensional @math -histogram and @math be an explicitly given @math -histogram. We want to correctly distinguish, with probability at least @math , between the case that @math versus @math . We design a computationally efficient algorithm for this hypothesis testing problem with sample complexity @math . Our algorithm is robust to model misspecification, i.e., succeeds even if @math is only promised to be close to a @math -histogram. Moreover, for @math , we show a nearly-matching sample complexity lower bound of @math when @math . Prior to our work, the sample complexity of the @math case was well-understood, but no algorithm with sub-learning sample complexity was known, even for @math . Our new upper and lower bounds have interesting conceptual implications regarding the relation between learning and testing in this setting.
The area of distribution inference under structural assumptions --- that is, inference about a distribution under the constraint that its probability density function satisfies certain qualitative properties --- is a classical topic in statistics starting with the pioneering work of Grenander @cite_9 on monotone distributions. The reader is referred to @cite_8 for a summary of the early work and to @cite_52 for a recent book on the subject. This topic is well-motivated in its own right, and has seen a recent surge of research activity in the statistics and econometrics communities, due to the ubiquity of structured distributions in the sciences. The conventional wisdom is that, under such structural constraints, the quality of the resulting estimators may dramatically improve, both in terms of sample size and in terms of computational efficiency.
@cite_9 @cite_52 @cite_8
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We show that a class of statistical properties of distributions, which includes such practically relevant properties as entropy, the number of distinct elements, and distance metrics between pairs of distributions, can be estimated given a sublinear sized sample. Specifically, given a sample consisting of independent draws from any distribution over at most k distinct elements, these properties can be estimated accurately using a sample of size O(k log k). For these estimation tasks, this performance is optimal, to constant factors. Complementing these theoretical results, we also demonstrate that our estimators perform exceptionally well, in practice, for a variety of estimation tasks, on a variety of natural distributions, for a wide range of parameters. The key step in our approach is to first use the sample to characterize the “unseen” portion of the distribution—effectively reconstructing this portion of the distribution as accurately as if one had a logarithmic factor larger sample. This goes beyond such tools as the Good-Turing frequency estimation scheme, which estimates the total probability mass of the unobserved portion of the distribution: We seek to estimate the shape of the unobserved portion of the distribution. This work can be seen as introducing a robust, general, and theoretically principled framework that, for many practical applications, essentially amplifies the sample size by a logarithmic factor; we expect that it may be fruitfully used as a component within larger machine learning and statistical analysis systems. We introduce and investigate a new model of learning probability distributions from independent draws. Our model is inspired by the popular Probably Approximately Correct (PAC) model for learning boolean functions from labeled examples [24], in the sense that we emphasize efficient and approximate learning, and we study the learnability of restricted classes of target distributions. The dist ribut ion classes we examine are often defined by some simple computational mechanism for transforming a truly random string of input bits (which is not visible to the learning algorithm) into the stochastic observation (output) seen by the learning algorithm. In this paper, we concentrate on discrete distributions over O, I n. The problem of inferring an approximation to an unknown probability distribution on the basis of independent draws has a long and complex history in the pattern recognition and statistics literature. For instance, the problem of estimating the parameters of a Gaussian density in highdimensional space is one of the most studied statistical problems. Distribution learning problems have often been investigated in the context of unsupervised learning, in which a linear mixture of two or more distributions is generating the observations, and the final goal is not to model the distributions themselves, but to predict from which distribution each observation was drawn. Data clustering methods are a common tool here. There is also a large literature on nonpararnetric density estimation, in which no assumptions are made on the unknown target density. Nearest-neighbor approaches to the unsupervised learning problem often arise in the nonparametric setting. While we obviously cannot do justice to these areas here, the books of Duda and Hart [9] and Vapnik [25] provide excellent overviews and introductions to the pattern recognition work, as well as many pointers for further reading. See also Izenman’s recent survey article [16]. Roughly speaking, our work departs from the traditional statistical and pattern recognition approaches in two ways. First, we place explicit emphasis on the comput ationrd complexity of distribution learning. It seems fair to say that while previous research has provided an excellent understanding of the information-theoretic issues involved in dis- We introduce a new approach to characterizing the unobserved portion of a distribution, which provides sublinear--sample estimators achieving arbitrarily small additive constant error for a class of properties that includes entropy and distribution support size. Additionally, we show new matching lower bounds. Together, this settles the longstanding question of the sample complexities of these estimation problems, up to constant factors. Our algorithm estimates these properties up to an arbitrarily small additive constant, using O(n log n) samples, where n is a bound on the support size, or in the case of estimating the support size, 1 n is a lower bound on the probability of any element of the domain. Previously, no explicit sublinear--sample algorithms for either of these problems were known. Our algorithm is also computationally extremely efficient, running in time linear in the number of samples used. In the second half of the paper, we provide a matching lower bound of Ω(n log n) samples for estimating entropy or distribution support size to within an additive constant. The previous lower-bounds on these sample complexities were n 2O(√log n). To show our lower bound, we prove two new and natural multivariate central limit theorems (CLTs); the first uses Stein's method to relate the sum of independent distributions to the multivariate Gaussian of corresponding mean and covariance, under the earthmover distance metric (also known as the Wasserstein metric). We leverage this central limit theorem to prove a stronger but more specific central limit theorem for "generalized multinomial" distributions---a large class of discrete distributions, parameterized by matrices, that represents sums of independent binomial or multinomial distributions, and describes many distributions encountered in computer science. Convergence here is in the strong sense of statistical distance, which immediately implies that any algorithm with input drawn from a generalized multinomial distribution behaves essentially as if the input were drawn from a discretized Gaussian with the same mean and covariance. Such tools in the multivariate setting are rare, and we hope this new tool will be of use to the community. We present a new solution to the ecological inference'' problem, of learning individual-level associations from aggregate data. This problem has a long history and has attracted much attention, debate, claims that it is unsolvable, and purported solutions. Unlike other ecological inference techniques, our method makes use of unlabeled individual-level data by embedding the distribution over these predictors into a vector in Hilbert space. Our approach relies on recent learning theory results for distribution regression, using kernel embeddings of distributions. Our novel approach to distribution regression exploits the connection between Gaussian process regression and kernel ridge regression, giving us a coherent, Bayesian approach to learning and inference and a convenient way to include prior information in the form of a spatial covariance function. Our approach is highly scalable as it relies on FastFood, a randomized explicit feature representation for kernel embeddings. We apply our approach to the challenging political science problem of modeling the voting behavior of demographic groups based on aggregate voting data. We consider the 2012 US Presidential election, and ask: what was the probability that members of various demographic groups supported Barack Obama, and how did this vary spatially across the country? Our results match standard survey-based exit polling data for the small number of states for which it is available, and serve to fill in the large gaps in this data, at a much higher degree of granularity.