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141
Clustering with Bregman Divergences
 JOURNAL OF MACHINE LEARNING RESEARCH
, 2005
"... A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergence ..."
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Cited by 310 (52 self)
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A wide variety of distortion functions are used for clustering, e.g., squared Euclidean distance, Mahalanobis distance and relative entropy. In this paper, we propose and analyze parametric hard and soft clustering algorithms based on a large class of distortion functions known as Bregman divergences. The proposed algorithms unify centroidbased parametric clustering approaches, such as classical kmeans and informationtheoretic clustering, which arise by special choices of the Bregman divergence. The algorithms maintain the simplicity and scalability of the classical kmeans algorithm, while generalizing the basic idea to a very large class of clustering loss functions. There are two main contributions in this paper. First, we pose the hard clustering problem in terms of minimizing the loss in Bregman information, a quantity motivated by ratedistortion theory, and present an algorithm to minimize this loss. Secondly, we show an explicit bijection between Bregman divergences and exponential families. The bijection enables the development of an alternative interpretation of an ecient EM scheme for learning models involving mixtures of exponential distributions. This leads to a simple soft clustering algorithm for all Bregman divergences.
Algebraic Algorithms for Sampling from Conditional Distributions
 Annals of Statistics
, 1995
"... We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so a ..."
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Cited by 192 (16 self)
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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.
A Guide to the Literature on Learning Probabilistic Networks From Data
, 1996
"... This literature review discusses different methods under the general rubric of learning Bayesian networks from data, and includes some overlapping work on more general probabilistic networks. Connections are drawn between the statistical, neural network, and uncertainty communities, and between the ..."
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Cited by 172 (0 self)
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This literature review discusses different methods under the general rubric of learning Bayesian networks from data, and includes some overlapping work on more general probabilistic networks. Connections are drawn between the statistical, neural network, and uncertainty communities, and between the different methodological communities, such as Bayesian, description length, and classical statistics. Basic concepts for learning and Bayesian networks are introduced and methods are then reviewed. Methods are discussed for learning parameters of a probabilistic network, for learning the structure, and for learning hidden variables. The presentation avoids formal definitions and theorems, as these are plentiful in the literature, and instead illustrates key concepts with simplified examples. Keywords Bayesian networks, graphical models, hidden variables, learning, learning structure, probabilistic networks, knowledge discovery. I. Introduction Probabilistic networks or probabilistic gra...
Relative Loss Bounds for Online Density Estimation with the Exponential Family of Distributions
 MACHINE LEARNING
, 2000
"... We consider online density estimation with a parameterized density from the exponential family. The online algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss, which is the n ..."
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Cited by 116 (11 self)
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We consider online density estimation with a parameterized density from the exponential family. The online algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss, which is the negative loglikelihood of the example with respect to the past parameter of the algorithm. An oline algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the online algorithm over the total loss of the best oline parameter. These relative loss bounds hold for an arbitrary sequence of examples. The goal is to design algorithms with the best possible relative loss bounds. We use a Bregman divergence to derive and analyze each algorithm. These divergences are relative entropies between two exponential distributions. We also use our methods to prove relative loss bounds for linear regression.
Accelerated training of conditional random fields with stochastic gradient methods
 In ICML
, 2006
"... We apply Stochastic MetaDescent (SMD), a stochastic gradient optimization method with gain vector adaptation, to the training of Conditional Random Fields (CRFs). On several large data sets, the resulting optimizer converges to the same quality of solution over an order of magnitude faster than lim ..."
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Cited by 95 (4 self)
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We apply Stochastic MetaDescent (SMD), a stochastic gradient optimization method with gain vector adaptation, to the training of Conditional Random Fields (CRFs). On several large data sets, the resulting optimizer converges to the same quality of solution over an order of magnitude faster than limitedmemory BFGS, the leading method reported to date. We report results for both exact and inexact inference techniques. 1.
Game Theory, Maximum Entropy, Minimum Discrepancy And Robust Bayesian Decision Theory
 ANNALS OF STATISTICS
, 2004
"... ..."
A scalable modular convex solver for regularized risk minimization
 In KDD. ACM
, 2007
"... A wide variety of machine learning problems can be described as minimizing a regularized risk functional, with different algorithms using different notions of risk and different regularizers. Examples include linear Support Vector Machines (SVMs), Logistic Regression, Conditional Random Fields (CRFs ..."
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Cited by 59 (14 self)
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A wide variety of machine learning problems can be described as minimizing a regularized risk functional, with different algorithms using different notions of risk and different regularizers. Examples include linear Support Vector Machines (SVMs), Logistic Regression, Conditional Random Fields (CRFs), and Lasso amongst others. This paper describes the theory and implementation of a highly scalable and modular convex solver which solves all these estimation problems. It can be parallelized on a cluster of workstations, allows for datalocality, and can deal with regularizers such as ℓ1 and ℓ2 penalties. At present, our solver implements 20 different estimation problems, can be easily extended, scales to millions of observations, and is up to 10 times faster than specialized solvers for many applications. The open source code is freely available as part of the ELEFANT toolbox.
Multiscale likelihood analysis and complexity penalized estimation
 Annals of Statistics
"... We describe here a framework for a certain class of multiscale likelihood factorizations wherein, in analogy to a wavelet decomposition of an L 2 function, a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and ..."
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Cited by 57 (20 self)
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We describe here a framework for a certain class of multiscale likelihood factorizations wherein, in analogy to a wavelet decomposition of an L 2 function, a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and the parameter vector localized in position and scale. The framework is developed as a set of sufficient conditions for the existence of such factorizations, formulated in analogy to those underlying a standard multiresolution analysis for wavelets, and hence can be viewed as a multiresolution analysis for likelihoods. We then consider the use of these factorizations in the task of nonparametric, complexity penalized likelihood estimation. We study the risk properties of certain thresholding and partitioning estimators, and demonstrate their adaptivity and nearoptimality, in a minimax sense over a broad range of function spaces, based on squared Hellinger distance as a loss function. In particular, our results provide an illustration of how properties of classical waveletbased estimators can be obtained in a single, unified framework
Additive Models, Boosting, and Inference for Generalized Divergences
 In Proc. 12th Annu. Conf. on Comput. Learning Theory
, 1999
"... We present a framework for designing incremental learning algorithms derived from generalized entropy functionals. Our approach is based on the use of Bregman divergences together with the associated class of additive models constructed using the Legendre transform. A particular oneparameter family ..."
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Cited by 39 (3 self)
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We present a framework for designing incremental learning algorithms derived from generalized entropy functionals. Our approach is based on the use of Bregman divergences together with the associated class of additive models constructed using the Legendre transform. A particular oneparameter family of Bregman divergences is shown to yield a family of loss functions that includes the loglikelihood criterion of logistic regression as a special case, and that closely approximates the exponential loss criterion used in the AdaBoost algorithms of Schapire et al., as the natural parameter of the family varies. We also show how the quadratic approximation of the gain in Bregman divergence results in a weighted leastsquares criterion. This leads to a family of incremental learning algorithms that builds upon and extends the recent interpretation of boosting in terms of additive models proposed by Friedman, Hastie, and Tibshirani. 1 Introduction Logistic regression is a widely used statisti...