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How to Use

First, initialize an RBM with the desired number of visible and hidden units.

rbm = RBM(num_visible = 6, num_hidden = 2)

Next, train the machine:

training_data = np.array([[1,1,1,0,0,0],[1,0,1,0,0,0],[1,1,1,0,0,0],[0,0,1,1,1,0], [0,0,1,1,0,0],[0,0,1,1,1,0]]) # A 6x6 matrix where each row is a training example and each column is a visible unit.
r.train(training_data, max_epochs = 5000) # Don't run the training for more than 5000 epochs.

Finally, run wild!

# Given a new set of visible units, we can see what hidden units are activated.
visible_data = np.array([[0,0,0,1,1,0]]) # A matrix with a single row that contains the states of the visible units. (We can also include more rows.)
r.run_visible(visible_data) # See what hidden units are activated.

# Given a set of hidden units, we can see what visible units are activated.
hidden_data = np.array([[1,0]]) # A matrix with a single row that contains the states of the hidden units. (We can also include more rows.)
r.run_hidden(hidden_data) # See what visible units are activated.

# We can let the network run freely (aka, daydream).
r.daydream(100) # Daydream for 100 steps on a single initialization.

Introduction

Suppose you ask a bunch of users to rate a set of movies on a 0-100 scale. In classical factor analysis, you could then try to explain each movie and user in terms of a set of latent factors. For example, movies like Star Wars and Lord of the Rings might have strong associations with a latent science fiction and fantasy factor, and users who like Wall-E and Toy Story might have strong associations with a latent Pixar factor.

Restricted Boltzmann Machines essentially perform a binary version of factor analysis. (This is one way of thinking about RBMs; there are, of course, others, and lots of different ways to use RBMs, but I'll adopt this approach for this post.) Instead of users rating a set of movies on a continuous scale, they simply tell you whether they like a movie or not, and the RBM will try to discover latent factors that can explain the activation of these movie choices.

More technically, a Restricted Boltzmann Machine is a stochastic neural network (neural network meaning we have neuron-like units whose binary activations depend on the neighbors they're connected to; stochastic meaning these activations have a probabilistic element) consisting of:

Furthermore, each visible unit is connected to all the hidden units (this connection is undirected, so each hidden unit is also connected to all the visible units), and the bias unit is connected to all the visible units and all the hidden units. To make learning easier, we restrict the network so that no visible unit is connected to any other visible unit and no hidden unit is connected to any other hidden unit.

For example, suppose we have a set of six movies (Harry Potter, Avatar, LOTR 3, Gladiator, Titanic, and Glitter) and we ask users to tell us which ones they want to watch. If we want to learn two latent units underlying movie preferences -- for example, two natural groups in our set of six movies appear to be SF/fantasy (containing Harry Potter, Avatar, and LOTR 3) and Oscar winners (containing LOTR 3, Gladiator, and Titanic), so we might hope that our latent units will correspond to these categories -- then our RBM would look like the following:

RBM Example

(Note the resemblance to a factor analysis graphical model.)

State Activation

Restricted Boltzmann Machines, and neural networks in general, work by updating the states of some neurons given the states of others, so let's talk about how the states of individual units change. Assuming we know the connection weights in our RBM (we'll explain how to learn these below), to update the state of unit $i$:

For example, let's suppose our two hidden units really do correspond to SF/fantasy and Oscar winners.

Learning Weights

So how do we learn the connection weights in our network? Suppose we have a bunch of training examples, where each training example is a binary vector with six elements corresponding to a user's movie preferences. Then for each epoch, do the following:

Continue until the network converges (i.e., the error between the training examples and their reconstructions falls below some threshold) or we reach some maximum number of epochs.

Why does this update rule make sense? Note that

So by adding $Positive(e_{ij}) - Negative(e_{ij})$ to each edge weight, we're helping the network's daydreams better match the reality of our training examples.

(You may hear this update rule called contrastive divergence, which is basically a funky term for "approximate gradient descent".)

Examples

I wrote a simple RBM implementation in Python (the code is heavily commented, so take a look if you're still a little fuzzy on how everything works), so let's use it to walk through some examples.

First, I trained the RBM using some fake data.

The network learned the following weights:

                 Bias Unit       Hidden 1        Hidden 2
Bias Unit       -0.08257658     -0.19041546      1.57007782 
Harry Potter    -0.82602559     -7.08986885      4.96606654 
Avatar          -1.84023877     -5.18354129      2.27197472 
LOTR 3           3.92321075      2.51720193      4.11061383 
Gladiator        0.10316995      6.74833901     -4.00505343 
Titanic         -0.97646029      3.25474524     -5.59606865 
Glitter         -4.44685751     -2.81563804     -2.91540988

Note that the first hidden unit seems to correspond to the Oscar winners, and the second hidden unit seems to correspond to the SF/fantasy movies, just as we were hoping.

What happens if we give the RBM a new user, George, who has (Harry Potter = 0, Avatar = 0, LOTR 3 = 0, Gladiator = 1, Titanic = 1, Glitter = 0) as his preferences? It turns the Oscar winners unit on (but not the SF/fantasy unit), correctly guessing that George probably likes movies that are Oscar winners.

What happens if we activate only the SF/fantasy unit, and run the RBM a bunch of different times? In my trials, it turned on Harry Potter, Avatar, and LOTR 3 three times; it turned on Avatar and LOTR 3, but not Harry Potter, once; and it turned on Harry Potter and LOTR 3, but not Avatar, twice. Note that, based on our training examples, these generated preferences do indeed match what we might expect real SF/fantasy fans want to watch.

Modifications

I tried to keep the connection-learning algorithm I described above pretty simple, so here are some modifications that often appear in practice:

rbmcmd

There is command-line tool to train and run RBM.

Here is the code that corresponds to the first example from "How to use" section

# First, initialize an RBM with the desired number of visible and hidden units.
./rbmcmd rbmstate.dat init  6  2  0.1

# Next, train the machine:
./rbmcmd rbmstate.dat train 5000 << \EOF
1 1 1 0 0 0
1 0 1 0 0 0
1 1 1 0 0 0
0 0 1 1 1 0
0 0 1 1 0 0
0 0 1 1 1 0
EOF

# Finally, run wild
# Given a new set of visible units, we can see what hidden units are activated.
echo "0 0 0 1 1 0" | ./rbmcmd rbmstate.dat run_visible
# 1 0

# Given a set of hidden units, we can see what visible units are activated.
echo "1 0" | ./rbmcmd rbmstate.dat run_hidden
# 0 1 1 1 0 0

# We can let the network run freely (aka, daydream).
# Daydream for 3 steps on a single initialization.
./rbmcmd rbmstate.dat daydream_trace 3 
# 0.901633539115 0.718084610948 0.00650400574634 0.853636318291 0.938241835347 0.0747538486547 
# 1.0 1.0 1.0 0.0 0.0 0.0 
# 1.0 1.0 1.0 0.0 0.0 0.0 


# See 5 dreams, each of 2 step from random data
./rbmcmd rbmstate.dat daydream 3 5
# 1 1 1 0 0 0 
# 0 0 1 1 0 0 
# 1 0 1 0 0 0 
# 1 0 1 0 0 0 
# 1 1 1 0 0 0 

Further

If you're interested in learning more about Restricted Boltzmann Machines, here are some good links.