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Information theory and learning: a physical approach | | Ilya Nemenman
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9 Sep 2000 | | Subject: | Data Analysis, Statistics and Probability; Disordered Systems and Neural Networks; Learning; Adaptation and Self-Organizing Systems | physics.data-an cond-mat.dis-nn cs.LG nlin.AO | | Abstract: | We try to establish a unified information theoretic approach to learning and to explore some of its applications. First, we define {em predictive information} as the mutual information between the past and the future of a time series, discuss its behavior as a function of the length of the series, and explain how other quantities of interest studied previously in learning theory - as well as in dynamical systems and statistical mechanics - emerge from this universally definable concept. We then prove that predictive information provides the {em unique measure for the complexity} of dynamics underlying the time series and show that there are classes of models characterized by {em power-law growth of the predictive information} that are qualitatively more complex than any of the systems that have been investigated before. Further, we investigate numerically the learning of a nonparametric probability density, which is an example of a problem with power-law complexity, and show that the proper Bayesian formulation of this problem provides for the `Occam’ factors that punish overly complex models and thus allow one {em to learn not only a solution within a specific model class, but also the class itself} using the data only and with very few a priori assumptions. We study a possible {em information theoretic method} that regularizes the learning of an undersampled discrete variable, and show that learning in such a setup goes through stages of very different complexities. Finally, we discuss how all of these ideas may be useful in various problems in physics, statistics, and, most importantly, biology. | | Source: | arXiv, physics/0009032 | | Services: | Forum | Review | PDF | Favorites | |
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