On the Convergence Properties of Contrastive Divergence

Contrastive Divergence (CD) is a popular method for estimating the parame- ters of Markov Random Fields (MRFs) by efficiently approxima ting an intractable term in the gradient of the MRF's log probability. Despite it s empirical success, basic theoretical questions on its convergence properties are currently open. In this paper, we analyze the CD1 update rule for Restricted Boltzmann Ma- chines (RBMs) with binary variables. We show that this update is not the gradient of any function, and we present an example of a somewhat contrived "regulariza- tion function" that causes the CD update to cycle indefinitel y. On the other hand, we prove that the CD update always has at least one fixed point w hen used with L2 regularization.