In [8]:

```
import matplotlib.cm as cm
import matplotlib.pyplot as plt
import gzip
import pickle
```

In [3]:

```
%matplotlib inline
```

In [9]:

```
with gzip.open('mnist.pkl.gz', 'rb') as f:
train_set, valid_set, test_set = pickle.load(f) #mnist.pkl is a tuple with length 3
```

In [10]:

```
train_x, train_y = train_set
```

In [11]:

```
train_x
```

Out[11]:

In [12]:

```
train_y
```

Out[12]:

In [13]:

```
len(train_y)
```

Out[13]:

In [16]:

```
len(train_x)
```

Out[16]:

In [17]:

```
len(train_x[0])
```

Out[17]:

由此可以看出，train_x是$50000\times 784$的矩阵，表明有50000个样本，每个样本长度为$784=28^2$，是像素尺度为$28\times 28$的图象。

In [18]:

```
plt.imshow(train_x[0].reshape((28, 28)), cmap = cm.Greys_r)
```

Out[18]:

In [20]:

```
from __future__ import print_function
import timeit
try:
import PIL.Image as Image
except ImportError:
import Image
import numpy
import theano
import theano.tensor as T
import os
from theano.sandbox.rng_mrg import MRG_RandomStreams as RandomStreams
from utils import tile_raster_images
from logistic_sgd import load_data
```

In [21]:

```
class RBM(object):
"""Restricted Boltzmann Machine (RBM) """
def __init__(
self,
input=None,
n_visible=784,
n_hidden=500,
W=None,
hbias=None,
vbias=None,
numpy_rng=None,
theano_rng=None
):
"""
RBM constructor. Defines the parameters of the model along with
basic operations for inferring hidden from visible (and vice-versa),
as well as for performing CD updates.
:param input: None for standalone RBMs or symbolic variable if RBM is
part of a larger graph.
:param n_visible: number of visible units
:param n_hidden: number of hidden units
:param W: None for standalone RBMs or symbolic variable pointing to a
shared weight matrix in case RBM is part of a DBN network; in a DBN,
the weights are shared between RBMs and layers of a MLP
:param hbias: None for standalone RBMs or symbolic variable pointing
to a shared hidden units bias vector in case RBM is part of a
different network
:param vbias: None for standalone RBMs or a symbolic variable
pointing to a shared visible units bias
"""
self.n_visible = n_visible
self.n_hidden = n_hidden
if numpy_rng is None:
# create a number generator
numpy_rng = numpy.random.RandomState(1234)
if theano_rng is None:
theano_rng = RandomStreams(numpy_rng.randint(2 ** 30))
if W is None:
# W is initialized with `initial_W` which is uniformely
# sampled from -4*sqrt(6./(n_visible+n_hidden)) and
# 4*sqrt(6./(n_hidden+n_visible)) the output of uniform if
# converted using asarray to dtype theano.config.floatX so
# that the code is runable on GPU
initial_W = numpy.asarray(
numpy_rng.uniform(
low=-4 * numpy.sqrt(6. / (n_hidden + n_visible)),
high=4 * numpy.sqrt(6. / (n_hidden + n_visible)),
size=(n_visible, n_hidden)
),
dtype=theano.config.floatX
)
# theano shared variables for weights and biases
W = theano.shared(value=initial_W, name='W', borrow=True)
if hbias is None:
# create shared variable for hidden units bias
hbias = theano.shared(
value=numpy.zeros(
n_hidden,
dtype=theano.config.floatX
),
name='hbias',
borrow=True
)
if vbias is None:
# create shared variable for visible units bias
vbias = theano.shared(
value=numpy.zeros(
n_visible,
dtype=theano.config.floatX
),
name='vbias',
borrow=True
)
# initialize input layer for standalone RBM or layer0 of DBN
self.input = input
if not input:
self.input = T.matrix('input')
self.W = W
self.hbias = hbias
self.vbias = vbias
self.theano_rng = theano_rng
# **** WARNING: It is not a good idea to put things in this list
# other than shared variables created in this function.
self.params = [self.W, self.hbias, self.vbias]
# end-snippet-1
def free_energy(self, v_sample):
''' Function to compute the free energy '''
wx_b = T.dot(v_sample, self.W) + self.hbias
vbias_term = T.dot(v_sample, self.vbias)
hidden_term = T.sum(T.log(1 + T.exp(wx_b)), axis=1)
return -hidden_term - vbias_term
def propup(self, vis):
'''This function propagates the visible units activation upwards to
the hidden units
Note that we return also the pre-sigmoid activation of the
layer. As it will turn out later, due to how Theano deals with
optimizations, this symbolic variable will be needed to write
down a more stable computational graph (see details in the
reconstruction cost function)
'''
pre_sigmoid_activation = T.dot(vis, self.W) + self.hbias
return [pre_sigmoid_activation, T.nnet.sigmoid(pre_sigmoid_activation)]
def sample_h_given_v(self, v0_sample):
''' This function infers state of hidden units given visible units '''
# compute the activation of the hidden units given a sample of
# the visibles
pre_sigmoid_h1, h1_mean = self.propup(v0_sample)
# get a sample of the hiddens given their activation
# Note that theano_rng.binomial returns a symbolic sample of dtype
# int64 by default. If we want to keep our computations in floatX
# for the GPU we need to specify to return the dtype floatX
h1_sample = self.theano_rng.binomial(size=h1_mean.shape,
n=1, p=h1_mean,
dtype=theano.config.floatX)
return [pre_sigmoid_h1, h1_mean, h1_sample]
def propdown(self, hid):
'''This function propagates the hidden units activation downwards to
the visible units
Note that we return also the pre_sigmoid_activation of the
layer. As it will turn out later, due to how Theano deals with
optimizations, this symbolic variable will be needed to write
down a more stable computational graph (see details in the
reconstruction cost function)
'''
pre_sigmoid_activation = T.dot(hid, self.W.T) + self.vbias
return [pre_sigmoid_activation, T.nnet.sigmoid(pre_sigmoid_activation)]
def sample_v_given_h(self, h0_sample):
''' This function infers state of visible units given hidden units '''
# compute the activation of the visible given the hidden sample
pre_sigmoid_v1, v1_mean = self.propdown(h0_sample)
# get a sample of the visible given their activation
# Note that theano_rng.binomial returns a symbolic sample of dtype
# int64 by default. If we want to keep our computations in floatX
# for the GPU we need to specify to return the dtype floatX
v1_sample = self.theano_rng.binomial(size=v1_mean.shape,
n=1, p=v1_mean,
dtype=theano.config.floatX)
return [pre_sigmoid_v1, v1_mean, v1_sample]
def gibbs_hvh(self, h0_sample):
''' This function implements one step of Gibbs sampling,
starting from the hidden state'''
pre_sigmoid_v1, v1_mean, v1_sample = self.sample_v_given_h(h0_sample)
pre_sigmoid_h1, h1_mean, h1_sample = self.sample_h_given_v(v1_sample)
return [pre_sigmoid_v1, v1_mean, v1_sample,
pre_sigmoid_h1, h1_mean, h1_sample]
def gibbs_vhv(self, v0_sample):
''' This function implements one step of Gibbs sampling,
starting from the visible state'''
pre_sigmoid_h1, h1_mean, h1_sample = self.sample_h_given_v(v0_sample)
pre_sigmoid_v1, v1_mean, v1_sample = self.sample_v_given_h(h1_sample)
return [pre_sigmoid_h1, h1_mean, h1_sample,
pre_sigmoid_v1, v1_mean, v1_sample]
# start-snippet-2
def get_cost_updates(self, lr=0.1, persistent=None, k=1):
"""This functions implements one step of CD-k or PCD-k
:param lr: learning rate used to train the RBM
:param persistent: None for CD. For PCD, shared variable
containing old state of Gibbs chain. This must be a shared
variable of size (batch size, number of hidden units).
:param k: number of Gibbs steps to do in CD-k/PCD-k
Returns a proxy for the cost and the updates dictionary. The
dictionary contains the update rules for weights and biases but
also an update of the shared variable used to store the persistent
chain, if one is used.
"""
# compute positive phase
pre_sigmoid_ph, ph_mean, ph_sample = self.sample_h_given_v(self.input)
# decide how to initialize persistent chain:
# for CD, we use the newly generate hidden sample
# for PCD, we initialize from the old state of the chain
if persistent is None:
chain_start = ph_sample
else:
chain_start = persistent
# end-snippet-2
# perform actual negative phase
# in order to implement CD-k/PCD-k we need to scan over the
# function that implements one gibbs step k times.
# Read Theano tutorial on scan for more information :
# http://deeplearning.net/software/theano/library/scan.html
# the scan will return the entire Gibbs chain
(
[
pre_sigmoid_nvs,
nv_means,
nv_samples,
pre_sigmoid_nhs,
nh_means,
nh_samples
],
updates
) = theano.scan(
self.gibbs_hvh,
# the None are place holders, saying that
# chain_start is the initial state corresponding to the
# 6th output
outputs_info=[None, None, None, None, None, chain_start],
n_steps=k,
name="gibbs_hvh"
)
# start-snippet-3
# determine gradients on RBM parameters
# note that we only need the sample at the end of the chain
chain_end = nv_samples[-1]
cost = T.mean(self.free_energy(self.input)) - T.mean(
self.free_energy(chain_end))
# We must not compute the gradient through the gibbs sampling
gparams = T.grad(cost, self.params, consider_constant=[chain_end])
# end-snippet-3 start-snippet-4
# constructs the update dictionary
for gparam, param in zip(gparams, self.params):
# make sure that the learning rate is of the right dtype
updates[param] = param - gparam * T.cast(
lr,
dtype=theano.config.floatX
)
if persistent:
# Note that this works only if persistent is a shared variable
updates[persistent] = nh_samples[-1]
# pseudo-likelihood is a better proxy for PCD
monitoring_cost = self.get_pseudo_likelihood_cost(updates)
else:
# reconstruction cross-entropy is a better proxy for CD
monitoring_cost = self.get_reconstruction_cost(updates,
pre_sigmoid_nvs[-1])
return monitoring_cost, updates
# end-snippet-4
def get_pseudo_likelihood_cost(self, updates):
"""Stochastic approximation to the pseudo-likelihood"""
# index of bit i in expression p(x_i | x_{\i})
bit_i_idx = theano.shared(value=0, name='bit_i_idx')
# binarize the input image by rounding to nearest integer
xi = T.round(self.input)
# calculate free energy for the given bit configuration
fe_xi = self.free_energy(xi)
# flip bit x_i of matrix xi and preserve all other bits x_{\i}
# Equivalent to xi[:,bit_i_idx] = 1-xi[:, bit_i_idx], but assigns
# the result to xi_flip, instead of working in place on xi.
xi_flip = T.set_subtensor(xi[:, bit_i_idx], 1 - xi[:, bit_i_idx])
# calculate free energy with bit flipped
fe_xi_flip = self.free_energy(xi_flip)
# equivalent to e^(-FE(x_i)) / (e^(-FE(x_i)) + e^(-FE(x_{\i})))
cost = T.mean(self.n_visible * T.log(T.nnet.sigmoid(fe_xi_flip -
fe_xi)))
# increment bit_i_idx % number as part of updates
updates[bit_i_idx] = (bit_i_idx + 1) % self.n_visible
return cost
def get_reconstruction_cost(self, updates, pre_sigmoid_nv):
"""Approximation to the reconstruction error
Note that this function requires the pre-sigmoid activation as
input. To understand why this is so you need to understand a
bit about how Theano works. Whenever you compile a Theano
function, the computational graph that you pass as input gets
optimized for speed and stability. This is done by changing
several parts of the subgraphs with others. One such
optimization expresses terms of the form log(sigmoid(x)) in
terms of softplus. We need this optimization for the
cross-entropy since sigmoid of numbers larger than 30. (or
even less then that) turn to 1. and numbers smaller than
-30. turn to 0 which in terms will force theano to compute
log(0) and therefore we will get either -inf or NaN as
cost. If the value is expressed in terms of softplus we do not
get this undesirable behaviour. This optimization usually
works fine, but here we have a special case. The sigmoid is
applied inside the scan op, while the log is
outside. Therefore Theano will only see log(scan(..)) instead
of log(sigmoid(..)) and will not apply the wanted
optimization. We can not go and replace the sigmoid in scan
with something else also, because this only needs to be done
on the last step. Therefore the easiest and more efficient way
is to get also the pre-sigmoid activation as an output of
scan, and apply both the log and sigmoid outside scan such
that Theano can catch and optimize the expression.
"""
cross_entropy = T.mean(
T.sum(
self.input * T.log(T.nnet.sigmoid(pre_sigmoid_nv)) +
(1 - self.input) * T.log(1 - T.nnet.sigmoid(pre_sigmoid_nv)),
axis=1
)
)
return cross_entropy
```

In [22]:

```
def test_rbm(learning_rate=0.1, training_epochs=15,
dataset='mnist.pkl.gz', batch_size=20,
n_chains=20, n_samples=10, output_folder='rbm_plots',
n_hidden=500):
"""
Demonstrate how to train and afterwards sample from it using Theano.
This is demonstrated on MNIST.
:param learning_rate: learning rate used for training the RBM
:param training_epochs: number of epochs used for training
:param dataset: path the the pickled dataset
:param batch_size: size of a batch used to train the RBM
:param n_chains: number of parallel Gibbs chains to be used for sampling
:param n_samples: number of samples to plot for each chain
"""
datasets = load_data(dataset)
train_set_x, train_set_y = datasets[0]
test_set_x, test_set_y = datasets[2]
# compute number of minibatches for training, validation and testing
n_train_batches = train_set_x.get_value(borrow=True).shape[0] // batch_size
# allocate symbolic variables for the data
index = T.lscalar() # index to a [mini]batch
x = T.matrix('x') # the data is presented as rasterized images
rng = numpy.random.RandomState(123)
theano_rng = RandomStreams(rng.randint(2 ** 30))
# initialize storage for the persistent chain (state = hidden
# layer of chain)
persistent_chain = theano.shared(numpy.zeros((batch_size, n_hidden),
dtype=theano.config.floatX),
borrow=True)
# construct the RBM class
rbm = RBM(input=x, n_visible=28 * 28,
n_hidden=n_hidden, numpy_rng=rng, theano_rng=theano_rng)
# get the cost and the gradient corresponding to one step of CD-15
cost, updates = rbm.get_cost_updates(lr=learning_rate,
persistent=persistent_chain, k=15)
#################################
# Training the RBM #
#################################
if not os.path.isdir(output_folder):
os.makedirs(output_folder)
os.chdir(output_folder)
# start-snippet-5
# it is ok for a theano function to have no output
# the purpose of train_rbm is solely to update the RBM parameters
train_rbm = theano.function(
[index],
cost,
updates=updates,
givens={
x: train_set_x[index * batch_size: (index + 1) * batch_size]
},
name='train_rbm'
)
plotting_time = 0.
start_time = timeit.default_timer()
# go through training epochs
for epoch in range(training_epochs):
# go through the training set
mean_cost = []
for batch_index in range(n_train_batches):
mean_cost += [train_rbm(batch_index)]
print('Training epoch %d, cost is ' % epoch, numpy.mean(mean_cost))
# Plot filters after each training epoch
plotting_start = timeit.default_timer()
# Construct image from the weight matrix
image = Image.fromarray(
tile_raster_images(
X=rbm.W.get_value(borrow=True).T,
img_shape=(28, 28),
tile_shape=(10, 10),
tile_spacing=(1, 1)
)
)
image.save('filters_at_epoch_%i.png' % epoch)
plotting_stop = timeit.default_timer()
plotting_time += (plotting_stop - plotting_start)
end_time = timeit.default_timer()
pretraining_time = (end_time - start_time) - plotting_time
print ('Training took %f minutes' % (pretraining_time / 60.))
# end-snippet-5 start-snippet-6
#################################
# Sampling from the RBM #
#################################
# find out the number of test samples
number_of_test_samples = test_set_x.get_value(borrow=True).shape[0]
# pick random test examples, with which to initialize the persistent chain
test_idx = rng.randint(number_of_test_samples - n_chains)
persistent_vis_chain = theano.shared(
numpy.asarray(
test_set_x.get_value(borrow=True)[test_idx:test_idx + n_chains],
dtype=theano.config.floatX
)
)
# end-snippet-6 start-snippet-7
plot_every = 1000
# define one step of Gibbs sampling (mf = mean-field) define a
# function that does `plot_every` steps before returning the
# sample for plotting
(
[
presig_hids,
hid_mfs,
hid_samples,
presig_vis,
vis_mfs,
vis_samples
],
updates
) = theano.scan(
rbm.gibbs_vhv,
outputs_info=[None, None, None, None, None, persistent_vis_chain],
n_steps=plot_every,
name="gibbs_vhv"
)
# add to updates the shared variable that takes care of our persistent
# chain :.
updates.update({persistent_vis_chain: vis_samples[-1]})
# construct the function that implements our persistent chain.
# we generate the "mean field" activations for plotting and the actual
# samples for reinitializing the state of our persistent chain
sample_fn = theano.function(
[],
[
vis_mfs[-1],
vis_samples[-1]
],
updates=updates,
name='sample_fn'
)
# create a space to store the image for plotting ( we need to leave
# room for the tile_spacing as well)
image_data = numpy.zeros(
(29 * n_samples + 1, 29 * n_chains - 1),
dtype='uint8'
)
for idx in range(n_samples):
# generate `plot_every` intermediate samples that we discard,
# because successive samples in the chain are too correlated
vis_mf, vis_sample = sample_fn()
print(' ... plotting sample %d' % idx)
image_data[29 * idx:29 * idx + 28, :] = tile_raster_images(
X=vis_mf,
img_shape=(28, 28),
tile_shape=(1, n_chains),
tile_spacing=(1, 1)
)
# construct image
image = Image.fromarray(image_data)
image.save('samples.png')
# end-snippet-7
os.chdir('../')
```

In [23]:

```
test_rbm()
```

In [1]:

```
from theano.sandbox.rng_mrg import MRG_RandomStreams as RandomStreams
```

In [4]:

```
import numpy
```

In [5]:

```
numpy_rng = numpy.random.RandomState(1234)
```

In [2]:

```
help(RandomStreams)
```

In [6]:

```
theano_rng = RandomStreams(numpy_rng.randint(2 ** 30))
```

In [7]:

```
theano_rng
```

Out[7]:

In [12]:

```
theano_rng.binomial(size=(100,10))# size is tupe of int or theano variables
```

Out[12]:

In [14]:

```
test = theano_rng.binomial(size=(100,10), p=0.4)
```

In [15]:

```
print(test)
```