On different facets of regularization theory
Neural Computation
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In learning problems available information is usually divided into two categories: examples of function values (or training data) and prior information (e.g. a smoothness constraint). This paper 1.) studies aspects on which these two categories usually differ, like their relevance for generalization and their role in the loss function, 2.) presents a unifying formalism, where both types of information are identified with answers to generalized questions, 3.) shows what kind of generalized information is necessary to enable learning, 4.) aims to put usual training data and prior information on a more equal footing by discussing possibilities and variants of measurement and control for generalized questions, including the examples of smoothness and symmetries, 5.) reviews shortly the measurement of linguistic concepts based on fuzzy priors, and principles to combine preprocessors, 6.) uses a Bayesian decision theoretic framework, contrasting parallel and inverse decision problems, 7.) proposes, for problems with non--approximation aspects, a Bayesian two step approximation consisting of posterior maximization and a subsequent risk minimization, 8.) analyses empirical risk minimization under the aspect of nonlocal information 9.) compares the Bayesian two step approximation with empirical risk minimization, including their interpretations of Occam''s razor, 10.) formulates examples of stationarity conditions for the maximum posterior approximation with nonlocal and nonconvex priors, leading to inhomogeneous nonlinear equations, similar for example to equations in scattering theory in physics. In summary, this paper focuses on the dependencies between answers to different questions. Because not training examples alone but such dependencies enable generalization, it emphasizes the need of their empirical measurement and control and of a more explicit treatment in theory.