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Semi-Supervised Learning$
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Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien

Print publication date: 2006

Print ISBN-13: 9780262033589

Published to MIT Press Scholarship Online: August 2013

DOI: 10.7551/mitpress/9780262033589.001.0001

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Entropy Regularization

Entropy Regularization

Chapter:
(p.151) 9 Entropy Regularization
Source:
Semi-Supervised Learning
Author(s):

Grandvalet Yves

Bengio Yoshua

Publisher:
The MIT Press
DOI:10.7551/mitpress/9780262033589.003.0009

This chapter promotes the use of entropy regularization as a means to benefit from unlabeled data in the framework of maximum a posteriori estimation. The learning criterion is derived from clearly stated assumptions and can be applied to any smoothly parameterized model of posterior probabilities. The regularization scheme favors low-density separation, without any modeling of the density of input features. The contribution of unlabeled data to the learning criterion induces local optima, but this problem can be alleviated by deterministic annealing. For well-behaved models of posterior probabilities, deterministic annealing expectation-maximization (EM) provides a decomposition of the learning problem in a series of concave subproblems. Other approaches to the semi-supervised problem are shown to be close relatives or limiting cases of entropy regularization. A series of experiments illustrates the good behavior of the algorithm in terms of performance and robustness with respect to the violation of the postulated low-density separation assumption.

Keywords:   maximum a posteriori estimation, entropy regularization, unlabeled data, posterior probabilities, regularization scheme, low-density separation, expectation-maximization, EM

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