Empirical Bayes Method for Boltzmann Machines
This work addresses a computational bottleneck for researchers using Boltzmann machines, but it is incremental due to the bias and limitations in the proposed solution.
The study tackled the computational challenge of estimating hyperparameters in Boltzmann machines using an empirical Bayes method, which involves intractable integrations, by proposing a fast algorithm based on the replica method and Plefka expansion, though it introduced bias with unnatural behavior related to dataset size.
In this study, we consider an empirical Bayes method for Boltzmann machines and propose an algorithm for it. The empirical Bayes method allows estimation of the values of the hyperparameters of the Boltzmann machine by maximizing a specific likelihood function referred to as the empirical Bayes likelihood function in this study. However, the maximization is computationally hard because the empirical Bayes likelihood function involves intractable integrations of the partition function. The proposed algorithm avoids this computational problem by using the replica method and the Plefka expansion. Our method does not require any iterative procedures and is quite simple and fast, though it introduces a bias to the estimate, which exhibits an unnatural behavior with respect to the size of the dataset. This peculiar behavior is supposed to be due to the approximate treatment by the Plefka expansion. A possible extension to overcome this behavior is also discussed.