2,946 research outputs found

    Estimating Mixture Entropy with Pairwise Distances

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    Mixture distributions arise in many parametric and non-parametric settings -- for example, in Gaussian mixture models and in non-parametric estimation. It is often necessary to compute the entropy of a mixture, but, in most cases, this quantity has no closed-form expression, making some form of approximation necessary. We propose a family of estimators based on a pairwise distance function between mixture components, and show that this estimator class has many attractive properties. For many distributions of interest, the proposed estimators are efficient to compute, differentiable in the mixture parameters, and become exact when the mixture components are clustered. We prove this family includes lower and upper bounds on the mixture entropy. The Chernoff α\alpha-divergence gives a lower bound when chosen as the distance function, with the Bhattacharyya distance providing the tightest lower bound for components that are symmetric and members of a location family. The Kullback-Leibler divergence gives an upper bound when used as the distance function. We provide closed-form expressions of these bounds for mixtures of Gaussians, and discuss their applications to the estimation of mutual information. We then demonstrate that our bounds are significantly tighter than well-known existing bounds using numeric simulations. This estimator class is very useful in optimization problems involving maximization/minimization of entropy and mutual information, such as MaxEnt and rate distortion problems.Comment: Corrects several errata in published version, in particular in Section V (bounds on mutual information

    Nonlinear Information Bottleneck

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    Information bottleneck (IB) is a technique for extracting information in one random variable XX that is relevant for predicting another random variable YY. IB works by encoding XX in a compressed "bottleneck" random variable MM from which YY can be accurately decoded. However, finding the optimal bottleneck variable involves a difficult optimization problem, which until recently has been considered for only two limited cases: discrete XX and YY with small state spaces, and continuous XX and YY with a Gaussian joint distribution (in which case optimal encoding and decoding maps are linear). We propose a method for performing IB on arbitrarily-distributed discrete and/or continuous XX and YY, while allowing for nonlinear encoding and decoding maps. Our approach relies on a novel non-parametric upper bound for mutual information. We describe how to implement our method using neural networks. We then show that it achieves better performance than the recently-proposed "variational IB" method on several real-world datasets

    Hierarchical relational models for document networks

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    We develop the relational topic model (RTM), a hierarchical model of both network structure and node attributes. We focus on document networks, where the attributes of each document are its words, that is, discrete observations taken from a fixed vocabulary. For each pair of documents, the RTM models their link as a binary random variable that is conditioned on their contents. The model can be used to summarize a network of documents, predict links between them, and predict words within them. We derive efficient inference and estimation algorithms based on variational methods that take advantage of sparsity and scale with the number of links. We evaluate the predictive performance of the RTM for large networks of scientific abstracts, web documents, and geographically tagged news.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS309 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

    A pragmatic look at deep imitation learning

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    The introduction of the generative adversarial imitation learning (GAIL) algorithm has spurred the development of scalable imitation learning approaches using deep neural networks. The GAIL objective can be thought of as 1) matching the expert policy's state distribution; 2) penalising the learned policy's state distribution; and 3) maximising entropy. While theoretically motivated, in practice GAIL can be difficult to apply, not least due to the instabilities of adversarial training. In this paper, we take a pragmatic look at GAIL and related imitation learning algorithms. We implement and automatically tune a range of algorithms in a unified experimental setup, presenting a fair evaluation between the competing methods. From our results, our primary recommendation is to consider non-adversarial methods. Furthermore, we discuss the common components of imitation learning objectives, and present promising avenues for future research
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