Convergence of Contrastive Divergence Algorithm in Exponential Family

This paper studies the convergence properties of contrastive divergence algorithm for parameter inference in exponential family, by relating it to Markov chain theory and stochastic stability literature. We prove that, under mild conditions and given a finite data sample $X_1,\dots,X_n \sim p_{\theta^*}$ i.i.d. in an event with probability approaching to 1, the sequence $\{\theta_t\}_{t \ge 0}$ generated by CD algorithm is a positive Harris recurrent chain, and thus processes an unique invariant distribution $\pi_n$. The invariant distribution concentrates around the Maximum Likelihood Estimate at a speed arbitrarily slower than $\sqrt{n}$, and the number of steps in Markov Chain Monte Carlo only affects the coefficient factor of the concentration rate. Finally we conclude that as $n \to \infty$, $$\limsup_{t \to \infty} \left\Vert \frac{1}{t} \sum_{s=1}^t \theta_s - \theta^*\right\Vert \overset{p}{\to} 0.$$