Copyright © 2006 The Institute of Electronics, Information and Communication Engineers
Regular Section-- Papers -- Algorithm Theory |
Generalization Performance of Subspace Bayes Approach in Linear Neural Networks
1 The authors are with Tokyo Institute of Technology, Yokohama-shi, 226–8503 Japan. E-mail: nakajima.s{at}cs.pi.titech.ac.jp, E-mail: swatanab{at}pi.titech.ac.jp, 2 The author is with Nikon Corporation, Kumagaya-shi, 360–8559 Japan.
In unidentifiable models, the Bayes estimation has the advantage of generalization performance over the maximum likelihood estimation. However, accurate approximation of the posterior distribution requires huge computational costs. In this paper, we consider an alternative approximation method, which we call a subspace Bayes approach. A subspace Bayes approach is an empirical Bayes approach where a part of the parameters are regarded as hyperparameters. Consequently, in some three-layer models, this approach requires much less computational costs than Markov chain Monte Carlo methods. We show that, in three-layer linear neural networks, a subspace Bayes approach is asymptotically equivalent to a positive-part James-Stein type shrinkage estimation, and theoretically clarify its generalization error and training error. We also discuss the domination over the maximum likelihood estimation and the relation to the variational Bayes approach.
Key Words: empirical Bayes, variational Bayes, neural networks, reduced-rank regression, James-Stein, unidentifiable
Manuscript received May 10, 2005. Manuscript revised August 24, 2005.