Generalization Performance of Subspace Bayes Approach in Linear Neural Networks

Shinichi NAKAJIMA  Sumio WATANABE  

Publication
IEICE TRANSACTIONS on Information and Systems   Vol.E89-D   No.3   pp.1128-1138
Publication Date: 2006/03/01
Online ISSN: 1745-1361
Print ISSN: 0916-8532
Type of Manuscript: PAPER
Category: Algorithm Theory
Keyword: 
empirical Bayes,  variational Bayes,  neural networks,  reduced-rank regression,  James-Stein,  unidentifiable,  

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Summary: 
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.