Analysis of Empirical MAP and Empirical Partially Bayes: Can They be Alternatives to Variational Bayes?

Variational Bayesian (VB) learning is known to be a promising approximation to Bayesian learning with computational efficiency. However, in some applications, e.g., large-scale collaborative filtering and tensor factorization, VB is still computationally too costly. In such cases, looser approximations such as MAP estimation and partially Bayesian (PB) learning, where a part of the parameters are point-estimated, seem attractive. In this paper, we theoretically investigate the behavior of the MAP and the PB solutions of matrix factorization. A notable finding is that the global solutions of MAP and PB in the empirical Bayesian scenario, where the hyperparameters are also estimated from observation, are trivial and useless, while their local solutions behave similarly to the global solution of VB. This suggests that empirical MAP and empirical PB with local search can be alternatives to empirical VB equipped with the useful automatic relevance determination property. Experiments support our theory.