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119
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 315 (50 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.
On the Generalization Ability of Online Learning Algorithms
 IEEE Transactions on Information Theory
, 2001
"... In this paper we show that online algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentrationofmeasure arguments and they hold for arbitrary onlin ..."
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Cited by 135 (8 self)
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In this paper we show that online algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentrationofmeasure arguments and they hold for arbitrary online learning algorithms. Furthermore, when applied to concrete online algorithms, our results yield tail bounds that in many cases are comparable or better than the best known bounds.
A generalization of principal component analysis to the exponential family
 Advances in Neural Information Processing Systems
, 2001
"... Principal component analysis (PCA) is a commonly applied technique for dimensionality reduction. PCA implicitly minimizes a squared loss function, which may be inappropriate for data that is not realvalued, such as binaryvalued data. This paper draws on ideas from the Exponential family, Generaliz ..."
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Cited by 112 (0 self)
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Principal component analysis (PCA) is a commonly applied technique for dimensionality reduction. PCA implicitly minimizes a squared loss function, which may be inappropriate for data that is not realvalued, such as binaryvalued data. This paper draws on ideas from the Exponential family, Generalized linear models, and Bregman distances, to give a generalization of PCA to loss functions that we argue are better suited to other data types. We describe algorithms for minimizing the loss functions, and give examples on simulated data. 1
General convergence results for linear discriminant updates
 Machine Learning
, 1997
"... Abstract. The problem of learning lineardiscriminant concepts can be solved by various mistakedriven update procedures, including the Winnow family of algorithms and the wellknown Perceptron algorithm. In this paper we define the general class of “quasiadditive ” algorithms, which includes Perce ..."
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Cited by 83 (0 self)
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Abstract. The problem of learning lineardiscriminant concepts can be solved by various mistakedriven update procedures, including the Winnow family of algorithms and the wellknown Perceptron algorithm. In this paper we define the general class of “quasiadditive ” algorithms, which includes Perceptron and Winnow as special cases. We give a single proof of convergence that covers a broad subset of algorithms in this class, including both Perceptron and Winnow, but also many new algorithms. Our proof hinges on analyzing a generic measure of progress construction that gives insight as to when and how such algorithms converge. Our measure of progress construction also permits us to obtain good mistake bounds for individual algorithms. We apply our unified analysis to new algorithms as well as existing algorithms. When applied to known algorithms, our method “automatically ” produces close variants of existing proofs (recovering similar bounds)—thus showing that, in a certain sense, these seemingly diverse results are fundamentally isomorphic. However, we also demonstrate that the unifying principles are more broadly applicable, and analyze a new class of algorithms that smoothly interpolate between the additiveupdate behavior of Perceptron and the multiplicativeupdate behavior of Winnow.
Incremental algorithms for hierarchical classification
 Journal of Machine Learning Research
, 2004
"... We study the problem of classifying data in a given taxonomy when classifications associated with multiple and/or partial paths are allowed. We introduce a new algorithm that incrementally learns a linearthreshold classifier for each node of the taxonomy. A hierarchical classification is obtained b ..."
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Cited by 81 (8 self)
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We study the problem of classifying data in a given taxonomy when classifications associated with multiple and/or partial paths are allowed. We introduce a new algorithm that incrementally learns a linearthreshold classifier for each node of the taxonomy. A hierarchical classification is obtained by evaluating the trained node classifiers in a topdown fashion. To evaluate classifiers in our multipath framework, we define a new hierarchical loss function, the Hloss, capturing the intuition that whenever a classification mistake is made on a node of the taxonomy, then no loss should be charged for any additional mistake occurring in the subtree of that node. Making no assumptions on the mechanism generating the data instances, and assuming a linear noise model for the labels, we bound the Hloss of our online algorithm in terms of the Hloss of a reference classifier knowing the true parameters of the labelgenerating process. We show that, in expectation, the excess cumulative Hloss grows at most logarithmically in the length of the data sequence. Furthermore, our analysis reveals the precise dependence of the rate of convergence on the eigenstructure of the data each node observes. Our theoretical results are complemented by a number of experiments on texual corpora. In these experiments we show that, after only one epoch of training, our algorithm performs much better than Perceptronbased hierarchical classifiers, and reasonably close to a hierarchical support vector machine.
Game Theory, Maximum Entropy, Minimum Discrepancy And Robust Bayesian Decision Theory
 ANNALS OF STATISTICS
, 2004
"... ..."
Competitive online statistics
 International Statistical Review
, 1999
"... A radically new approach to statistical modelling, which combines mathematical techniques of Bayesian statistics with the philosophy of the theory of competitive online algorithms, has arisen over the last decade in computer science (to a large degree, under the influence of Dawid’s prequential sta ..."
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Cited by 65 (10 self)
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A radically new approach to statistical modelling, which combines mathematical techniques of Bayesian statistics with the philosophy of the theory of competitive online algorithms, has arisen over the last decade in computer science (to a large degree, under the influence of Dawid’s prequential statistics). In this approach, which we call “competitive online statistics”, it is not assumed that data are generated by some stochastic mechanism; the bounds derived for the performance of competitive online statistical procedures are guaranteed to hold (and not just hold with high probability or on the average). This paper reviews some results in this area; the new material in it includes the proofs for the performance of the Aggregating Algorithm in the problem of linear regression with square loss. Keywords: Bayes’s rule, competitive online algorithms, linear regression, prequential statistics, worstcase analysis.
Adaptive and SelfConfident OnLine Learning Algorithms
, 2000
"... We study online learning in the linear regression framework. Most of the performance bounds for online algorithms in this framework assume a constant learning rate. To achieve these bounds the learning rate must be optimized based on a posteriori information. This information depends on the wh ..."
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Cited by 64 (7 self)
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We study online learning in the linear regression framework. Most of the performance bounds for online algorithms in this framework assume a constant learning rate. To achieve these bounds the learning rate must be optimized based on a posteriori information. This information depends on the whole sequence of examples and thus it is not available to any strictly online algorithm. We introduce new techniques for adaptively tuning the learning rate as the data sequence is progressively revealed. Our techniques allow us to prove essentially the same bounds as if we knew the optimal learning rate in advance. Moreover, such techniques apply to a wide class of online algorithms, including pnorm algorithms for generalized linear regression and Weighted Majority for linear regression with absolute loss. Our adaptive tunings are radically dierent from previous techniques, such as the socalled doubling trick. Whereas the doubling trick restarts the online algorithm several ti...
Relational Learning via Collective Matrix Factorization
, 2008
"... Relational learning is concerned with predicting unknown values of a relation, given a database of entities and observed relations among entities. An example of relational learning is movie rating prediction, where entities could include users, movies, genres, and actors. Relations would then encode ..."
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Cited by 62 (3 self)
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Relational learning is concerned with predicting unknown values of a relation, given a database of entities and observed relations among entities. An example of relational learning is movie rating prediction, where entities could include users, movies, genres, and actors. Relations would then encode users ’ ratings of movies, movies ’ genres, and actors ’ roles in movies. A common prediction technique given one pairwise relation, for example a #users × #movies ratings matrix, is lowrank matrix factorization. In domains with multiple relations, represented as multiple matrices, we may improve predictive accuracy by exploiting information from one relation while predicting another. To this end, we propose a collective matrix factorization model: we simultaneously factor several matrices, sharing parameters among factors when an entity participates in multiple relations. Each relation can have a different value type and error distribution; so, we allow nonlinear relationships between the parameters and outputs, using Bregman divergences to measure error. We extend standard alternating projection algorithms to our model, and derive an efficient Newton update for the projection. Furthermore, we propose stochastic optimization methods to deal with large, sparse matrices. Our model generalizes several existing matrix factorization methods, and therefore yields new largescale optimization algorithms for these problems. Our model can handle any pairwise relational schema and a
A secondorder perceptron algorithm
, 2005
"... Kernelbased linearthreshold algorithms, such as support vector machines and Perceptronlike algorithms, are among the best available techniques for solving pattern classification problems. In this paper, we describe an extension of the classical Perceptron algorithm, called secondorder Perceptr ..."
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Cited by 59 (20 self)
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Kernelbased linearthreshold algorithms, such as support vector machines and Perceptronlike algorithms, are among the best available techniques for solving pattern classification problems. In this paper, we describe an extension of the classical Perceptron algorithm, called secondorder Perceptron, and analyze its performance within the mistake bound model of online learning. The bound achieved by our algorithm depends on the sensitivity to secondorder data information and is the best known mistake bound for (efficient) kernelbased linearthreshold classifiers to date. This mistake bound, which strictly generalizes the wellknown Perceptron bound, is expressed in terms of the eigenvalues of the empirical data correlation matrix and depends on a parameter controlling the sensitivity of the algorithm to the distribution of these eigenvalues. Since the optimal setting of this parameter is not known a priori, we also analyze two variants of the secondorder Perceptron algorithm: one that adaptively sets the value of the parameter in terms of the number of mistakes made so far, and one that is parameterless, based on pseudoinverses.