I have recently implemented the variational autoencoder proposed in Kingma and Welling (2014)^{1} . This is a rather interesting unsupervised learning model. In contrast to most earlier work, Kingma and Welling (2014)^{1} optimize the variational lower bound directly using gradient ascent. What makes this possible is a trick they call the reparameterization trick. I am hoping to write another post later on what this trick is and how it helps in reducing the variance of the gradient estimates. Here I just wanted to share my implementation. This model is surprisingly easy to implement. Below is an implementation in theano with a replication of Figure4b from their paper.

"""
A Theano implementation of the variational autoencoder proposed in
Kingma, D. P., & Welling, M. (2014). Auto-Encoding Variational Bayes.
arXiv:1312.6114
28 May 2016
goker erdogan
https://github.com/gokererdogan
"""
import cPickle as pkl
import gzip
import numpy as np
import theano
import theano.tensor as T
from theano.tensor.shared_randomstreams import RandomStreams
from DeepLearning.common import load_mnist
# functions used for initializing weights and biases
def init_weight ( * shape ):
return 0.01 * np . random . randn ( * shape )
def init_bias ( * shape ):
return 0.01 * np . random . randn ( * shape )
if __name__ == "__main__" :
# BUILD THE MODEL
# random number generator used for sampling latent variables
srng = RandomStreams ( seed = 123 )
# model parameters
minibatch_size = 100
input_dim = 784
# number of hidden units in encoder (x -> z) network
encoder_hidden_dim = 500
# number of hidden units in decoder (z -> x) network
decoder_hidden_dim = 500
# number of latent variables
latent_dim = 2 # pairs of mu and sigma
# input to the network
x = T . fmatrix ( name = 'x' )
# Encoder network hidden layer
w1 = theano . shared ( init_weight ( input_dim , encoder_hidden_dim ), name = 'w1' )
b1 = theano . shared ( init_bias ( encoder_hidden_dim ), name = 'b1' )
h1 = T . tanh ( T . dot ( x , w1 ) + b1 )
# Encoder network outputs: means and standard deviations of the Gaussian distribution
# p(z|mu, sigma)
w_mu = theano . shared ( init_weight ( encoder_hidden_dim , latent_dim ), name = 'w_mu' )
b_mu = theano . shared ( init_bias ( latent_dim ), name = 'b_mu' )
mu = T . dot ( h1 , w_mu ) + b_mu
w_sigma = theano . shared ( init_weight ( encoder_hidden_dim , latent_dim ), name = 'w_sigma' )
b_sigma = theano . shared ( init_bias ( latent_dim ), name = 'b_sigma' )
sigma = T . dot ( h1 , w_sigma ) + b_sigma
# Latent variables
# Sample z from Normal(z|mu, sigma) using the reparameterization trick.
eps = srng . normal (( minibatch_size , latent_dim )) # draw random normal values
z = mu + ( eps * T . exp ( sigma )) # calculate latent variables
# Decoder network hidden layer
w2 = theano . shared ( init_weight ( latent_dim , decoder_hidden_dim ), name = 'w2' )
b2 = theano . shared ( init_bias ( decoder_hidden_dim ), name = 'b2' )
h2 = T . tanh ( T . dot ( z , w2 ) + b2 )
# Decoder network output
w3 = theano . shared ( init_weight ( decoder_hidden_dim , input_dim ), name = 'w3' )
b3 = theano . shared ( init_bias ( input_dim ), name = 'b3' )
y = T . nnet . sigmoid ( T . dot ( h2 , w3 ) + b3 )
# LOSS FUNCTION
# first KL term in Eqn. 10 in the paper. Acts as a regularization on latent variables.
ll_bound_kl_term = 0.5 * ( 1. + T . sum ( 2. * sigma ) -
T . sum ( T . square ( mu )) -
T . sum ( T . exp ( 2. * sigma ))) / minibatch_size
# second term in Eqn. 10, measures data fit.
# Note because the outputs are 0-1, we use binary crossentropy.
ll_bound_fit_term = T . sum ( - 1. * T . nnet . binary_crossentropy ( y , x )) / minibatch_size
# this is the log likelihood lower bound that we want to maximize
ll_bound = ll_bound_kl_term + ll_bound_fit_term
# L2 regularization
# We add L2 regularization on model parameters. This can be achieved by adding
# the Frobenius norm of each model parameter to loss function.
# L2 regularization weight
lamda = 0.01
# cost = -log ll bound + (lambda * L2 regularizer)
cost = - ll_bound + lamda * ( T . sum ( w1 ** 2 ) + T . sum ( b1 ** 2 ) + T . sum ( w_mu ** 2 ) +
T . sum ( b_mu ** 2 ) + T . sum ( w_sigma ** 2 ) +
T . sum ( b_sigma ** 2 ) + T . sum ( w2 ** 2 ) +
T . sum ( b2 ** 2 ) + T . sum ( w3 ** 2 ) +
T . sum ( b3 ** 2 ))
# theano function that calculates the gradient of the cost wrt to model parameters
dw1 , db1 , dw_mu , db_mu , dw_sigma , db_sigma , dw2 , db2 , dw3 , db3 = T . grad ( cost ,
[ w1 , b1 , w_mu ,
b_mu , w_sigma ,
b_sigma , w2 ,
b2 , w3 , b3 ])
# theano function for getting the output of the model for a given input
predict = theano . function ([ x ], y )
# TRAINING
learning_rate = 0.001 # my experience shows that larger learning rates lead to divergence
# number of training epochs, i.e., passes over training set.
# there are 50.000 samples in training set. 200 epochs means training over
# 10.000.000 samples. In the paper, they keep training for much longer (around 100.000.000)
epoch_count = 500
# report log_likelihood bound on training after this many training samples
report_interval = 20000
# load the training data
( tx , ty ), ( vx , vy ), ( _ , _ ) = load_mnist ( path = '../datasets' )
train_n = tx . shape [ 0 ]
val_n = vx . shape [ 0 ]
# we load the data into shared variables, this is recommended if you do training on GPU.
train_x = theano . shared ( np . asarray ( tx , dtype = np . float32 ))
val_x = theano . shared ( np . asarray ( vx , dtype = np . float32 ))
# index of the first sample in batch
batch_start = T . lscalar ()
# Theano function for training
# This function feeds the current batch to model and applies
# simple gradient descent updates on model parameters
train_model = theano . function ([ batch_start ], ll_bound ,
updates = (( w1 , w1 - learning_rate * dw1 ),
( b1 , b1 - learning_rate * db1 ),
( w_mu , w_mu - learning_rate * dw_mu ),
( b_mu , b_mu - learning_rate * db_mu ),
( w_sigma , w_sigma - learning_rate * dw_sigma ),
( b_sigma , b_sigma - learning_rate * db_sigma ),
( w2 , w2 - learning_rate * dw2 ),
( b2 , b2 - learning_rate * db2 ),
( w3 , w3 - learning_rate * dw3 ),
( b3 , b3 - learning_rate * db3 )),
givens = { x : train_x [ batch_start :( batch_start + minibatch_size )]})
# Theano function for calculating log ll bound on validation set
validate_model = theano . function ([ batch_start ], ll_bound ,
givens = { x : val_x [ batch_start :( batch_start + minibatch_size )]})
# arrays for keeping track of log ll bounds on training and validation sets
train_ll = []
val_ll = []
# TRAINING LOOP
for e in range ( epoch_count ):
ll = 0.0
for i in range ( 0 , train_n , minibatch_size ):
cll = train_model ( i ) # one step of gradient descent
ll += cll
# report log ll bound
if ( i + minibatch_size ) % report_interval == 0 :
ll = ( ll * minibatch_size ) / report_interval
print "|Epoch {0:d}| \t Train ll: {1:f}" . format ( e , ll )
train_ll . append ( ll )
ll = 0.0
# epoch over, shuffle training data
train_x = theano . shared ( np . asarray ( tx [ np . random . permutation ( train_n )], dtype = np . float32 ))
# calculate log ll on validation and report it
ll = 0.0
for i in range ( 0 , val_n , minibatch_size ):
cll = validate_model ( i )
ll += cll
ll = ( ll * minibatch_size ) / val_n
print "|Epoch {0:d}| \t Val ll: {1:f}" . format ( e , ll )
val_ll . append ( ll )
print ( "Training over." )
if latent_dim == 2 :
print ( "Generating latent space figure" )
img = np . zeros (( 28 * 20 , 28 * 20 ))
# function for calculating x given z
generate_fn = theano . function ([ z ], y )
# calculate min and max latent variable values over validation set
calc_latent = theano . function ([ batch_start ], z ,
givens = { x : val_x [ batch_start :( batch_start + minibatch_size )]})
minz = np . inf
maxz = - np . inf
for i in range ( 0 , val_n , minibatch_size ):
val_z = calc_latent ( i )
if np . min ( val_z ) < minz :
minz = np . min ( val_z )
if np . max ( val_z ) > maxz :
maxz = np . max ( val_z )
zvals = np . linspace ( minz , maxz , 20 )
for i , z1 in enumerate ( zvals ):
for j , z2 in enumerate ( zvals ):
xp = generate_fn ([ np . asarray ([ z1 , z2 ], dtype = np . float32 )])
img [( i * 28 ):(( i + 1 ) * 28 ), ( j * 28 ):(( j + 1 ) * 28 )] = xp . reshape (( 28 , 28 ))
"""
import matplotlib.pyplot as plt
plt.style.use('classic')
plt.imshow(img, cmap='gray')
"""
from scipy import misc
misc . imsave ( 'Fig4b.png' , img )