Contrastive Divergence Learning is a Time Reversal Adversarial Game
This work provides a foundational theoretical understanding of Contrastive Divergence learning for researchers and practitioners using unnormalized statistical models, clarifying its true objective and convergence properties.
The paper re-derives Contrastive Divergence (CD) learning without approximations, revealing it as an adversarial game where a discriminator identifies time-reversed Markov chains generated by the model. This clarifies CD's objective and its relationship to GANs, despite predating them.
Contrastive divergence (CD) learning is a classical method for fitting unnormalized statistical models to data samples. Despite its wide-spread use, the convergence properties of this algorithm are still not well understood. The main source of difficulty is an unjustified approximation which has been used to derive the gradient of the loss. In this paper, we present an alternative derivation of CD that does not require any approximation and sheds new light on the objective that is actually being optimized by the algorithm. Specifically, we show that CD is an adversarial learning procedure, where a discriminator attempts to classify whether a Markov chain generated from the model has been time-reversed. Thus, although predating generative adversarial networks (GANs) by more than a decade, CD is, in fact, closely related to these techniques. Our derivation settles well with previous observations, which have concluded that CD's update steps cannot be expressed as the gradients of any fixed objective function. In addition, as a byproduct, our derivation reveals a simple correction that can be used as an alternative to Metropolis-Hastings rejection, which is required when the underlying Markov chain is inexact (e.g. when using Langevin dynamics with a large step).