Restricted Boltzmann machines - update rule

Question

I am trying to wrap my head around Geoff Hinton's RBM tutoral on http://www.cs.toronto.edu/~hinton/absps/guideTR.pdf

I understand that the log gradient update rule requires us to take an unbiased sample from the data (e.g. a random training sample) and an unbiased sample from the model. To obtain an unbiased sample from the model - one can perform Gibbs sampling until convergence.

Here i am getting confused. If one performs Gibbs sampling until convergence - has one not already found the equilibrium of the RBM, and hence the update becomes irrelevant? Please point out my lack of understanding and conceptual error.

Answer 1

Assume you initialize an RBM with random weights. This RBM will now already represent some random probability distribution. The expectation $\langle v_i h_j \rangle_{\text{model}}$ of this random model can be estimated by running the Gibbs sampling until convergence. This "equilibrium of the RBM" as you call it, does not depend on the data yet, so it is just random. The goal of your training is to change the weights and biases in a way, that this $\langle v_i h_j \rangle_{\text{model}}$ fits to your data, i.e. $\langle v_i h_j \rangle_{\text{data}}$.

To do that, you make many small updates, which shift $\langle v_i h_j \rangle_{\text{model}}$ more and more towards $\langle v_i h_j \rangle_{\text{data}}$. This is exactly the update rule

$\Delta w_{ij} = \epsilon (\langle v_i h_j \rangle_{\text{data}}-\langle v_i h_j \rangle_{\text{model}})$

Now, the point I think you misunderstood: for each update (training step), you need to run the Gibbs chain to its equilibrium to get the current $\langle v_i h_j \rangle_{\text{model}}$. Then you update and get a better model. You repeat this, until your model fits the data well enough.

Of course you can't run this Gibbs chain until infinity, so Hinton proposed to run the chain for only one step.

Does this help to clarify?