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Why are the parameters of a Restricted Boltzmann machine trained for a fixed number of iterations (epochs) in many papers instead of choosing the ones corresponding to a stationary point of the likelihood?

Denote the observable data by $x$, hidden data by $h$, the energy function by $E$ and the normalizing constant by $Z$. The probability of $x$ is:
beginequation
P(x) = sum_h P(x,h) = sum_h frace^-E(x,h)Z.
endequation
The goal is to maximize the probability of $x$ conditional on the parameters of the model $theta$. Suppose one has access to a sample of $N$ observations of $x$ with typical element $x_i$. As estimator, one could find the roots of the derivative of the average sample log-likelihood:
beginequation
leftlbrace hattheta in hatTheta : N^-1 sum_x_i fracpartial log p(x_i) partial theta = 0 rightrbrace
endequation
and chose the one $theta^star in hatTheta $ maximizing the empirical likelihood. There exists many different ways to approximate the derivative of the log-likelihood to facilitate (maybe even permit) its computation. For example, Contrastive Divergence and Persistent Contrastive Divergence are used often. I wonder whether it makes sense to estimate the parameters $theta$ recursively until convergence while continuing to approximate the derivative of the log-likelihood. One could update the parameters after seeing each data point $x_i$ as:
beginequation
theta_i+1 = theta_i - eta_i fracpartial log p(x_i)partial theta_i
endequation
The practice I learned in Hinton et al. (2006) and in Tieleman (2008) is different though: Both papers define a number of fixed iterations a priori. Could somebody kindly educate me why recursively updating the parameters until convergence is not a good idea? In particular, I'm interested whether there's a theoretical flaw in my reasoning or whether computational capacities dictate sticking to a fixed number of iterations. I am grateful for any help!

I believe the problem is that log-likelihood is not directly computable, because of exponential in number of units complexity. There exist different proxies to the true log-likelihood, for instance pseudo log-likelihood (more here), and in principle you can train RBM until PLL is not changed too much.

Hovewer, there is a lot of randomness involved in the training process, so PLL most likely will be quite noisy (even true log-likelihood would be I believe).