Generalization error of linear neural networks in an empirical bayes approach
IJCAI'05 Proceedings of the 19th international joint conference on Artificial intelligence
Analytic solution of hierarchical variational bayes in linear inverse problem
ICANN'06 Proceedings of the 16th international conference on Artificial Neural Networks - Volume Part II
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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.