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تسجيل مستخدم جديدConvergence of Contrastive Divergence with Annealed Learning Rate in Exponential Family
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In our recent paper, we showed that in exponential family, contrastive divergence (CD) with fixed learning rate will give asymptotically consistent estimates cite{wu2016convergence}. In this paper, we establish consistency and convergence rate of CD with annealed learning rate $eta_t$. Specifically, suppose CD-$m$ generates the sequence of parameters ${theta_t}_{t ge 0}$ using an i.i.d. data sample $mathbf{X}_1^n sim p_{theta^*}$ of size $n$, then $delta_n(mathbf{X}_1^n) = limsup_{t to infty} Vert sum_{s=t_0}^t eta_s theta_s / sum_{s=t_0}^t eta_s - theta^* Vert$ converges in probability to 0 at a rate of $1/sqrt[3]{n}$. The number ($m$) of MCMC transitions in CD only affects the coefficient factor of convergence rate. Our proof is not a simple extension of the one in cite{wu2016convergence}. which depends critically on the fact that ${theta_t}_{t ge 0}$ is a homogeneous Markov chain conditional on the observed sample $mathbf{X}_1^n$. Under annealed learning rate, the homogeneous Markov property is not available and we have to develop an alternative approach based on super-martingales. Experiment results of CD on a fully-visible $2times 2$ Boltzmann Machine are provided to demonstrate our theoretical results.
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