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"""
This tutorial introduces logistic regression using Theano and conjugate
gradient descent.
Logistic regression is a probabilistic, linear classifier. It is parametrized
by a weight matrix :math:`W` and a bias vector :math:`b`. Classification is
done by projecting data points onto a set of hyperplanes, the distance to
which is used to determine a class membership probability.
Mathematically, this can be written as:
.. math::
P(Y=i|x, W,b) &= softmax_i(W x + b) \\
&= \frac {e^{W_i x + b_i}} {\sum_j e^{W_j x + b_j}}
The output of the model or prediction is then done by taking the argmax of
the vector whose i'th element is P(Y=i|x).
.. math::
y_{pred} = argmax_i P(Y=i|x,W,b)
This tutorial presents a conjugate gradient optimization method that is
suitable for smaller datasets.
References:
- textbooks: "Pattern Recognition and Machine Learning" -
Christopher M. Bishop, section 4.3.2
"""
__docformat__ = 'restructedtext en'
import os
import sys
import timeit
import numpy
import theano
import theano.tensor as T
from logistic_sgd import load_data
class LogisticRegression(object):
"""Multi-class Logistic Regression Class
The logistic regression is fully described by a weight matrix :math:`W`
and bias vector :math:`b`. Classification is done by projecting data
points onto a set of hyperplanes, the distance to which is used to
determine a class membership probability.
"""
def __init__(self, input, n_in, n_out):
""" Initialize the parameters of the logistic regression
:type input: theano.tensor.TensorType
:param input: symbolic variable that describes the input of the
architecture ( one minibatch)
:type n_in: int
:param n_in: number of input units, the dimension of the space in
which the datapoint lies
:type n_out: int
:param n_out: number of output units, the dimension of the space in
which the target lies
"""
# initialize theta = (W,b) with 0s; W gets the shape (n_in, n_out),
# while b is a vector of n_out elements, making theta a vector of
# n_in*n_out + n_out elements
self.theta = theano.shared(
value=numpy.zeros(
n_in * n_out + n_out,
dtype=theano.config.floatX
),
name='theta',
borrow=True
)
# W is represented by the fisr n_in*n_out elements of theta
self.W = self.theta[0:n_in * n_out].reshape((n_in, n_out))
# b is the rest (last n_out elements)
self.b = self.theta[n_in * n_out:n_in * n_out + n_out]
# compute vector of class-membership probabilities in symbolic form
self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W) + self.b)
# compute prediction as class whose probability is maximal in
# symbolic form
self.y_pred = T.argmax(self.p_y_given_x, axis=1)
# keep track of model input
self.input = input
def negative_log_likelihood(self, y):
"""Return the negative log-likelihood of the prediction of this model
under a given target distribution.
.. math::
\frac{1}{|\mathcal{D}|}\mathcal{L} (\theta=\{W,b\}, \mathcal{D}) =
\frac{1}{|\mathcal{D}|}\sum_{i=0}^{|\mathcal{D}|}
\log(P(Y=y^{(i)}|x^{(i)}, W,b)) \\
\ell (\theta=\{W,b\}, \mathcal{D})
:type y: theano.tensor.TensorType
:param y: corresponds to a vector that gives for each example the
correct label
"""
return -T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]), y])
def errors(self, y):
"""Return a float representing the number of errors in the minibatch
over the total number of examples of the minibatch
:type y: theano.tensor.TensorType
:param y: corresponds to a vector that gives for each example
the correct label
"""
# check if y has same dimension of y_pred
if y.ndim != self.y_pred.ndim:
raise TypeError(
'y should have the same shape as self.y_pred',
('y', y.type, 'y_pred', self.y_pred.type)
)
# check if y is of the correct datatype
if y.dtype.startswith('int'):
# the T.neq operator returns a vector of 0s and 1s, where 1
# represents a mistake in prediction
return T.mean(T.neq(self.y_pred, y))
else:
raise NotImplementedError()
def cg_optimization_mnist(n_epochs=50, mnist_pkl_gz='mnist.pkl.gz'):
"""Demonstrate conjugate gradient optimization of a log-linear model
This is demonstrated on MNIST.
:type n_epochs: int
:param n_epochs: number of epochs to run the optimizer
:type mnist_pkl_gz: string
:param mnist_pkl_gz: the path of the mnist training file from
http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz
"""
#############
# LOAD DATA #
#############
datasets = load_data(mnist_pkl_gz)
train_set_x, train_set_y = datasets[0]
valid_set_x, valid_set_y = datasets[1]
test_set_x, test_set_y = datasets[2]
batch_size = 600 # size of the minibatch
n_train_batches = train_set_x.get_value(borrow=True).shape[0] / batch_size
n_valid_batches = valid_set_x.get_value(borrow=True).shape[0] / batch_size
n_test_batches = test_set_x.get_value(borrow=True).shape[0] / batch_size
n_in = 28 * 28 # number of input units
n_out = 10 # number of output units
######################
# BUILD ACTUAL MODEL #
######################
print '... building the model'
# allocate symbolic variables for the data
minibatch_offset = T.lscalar() # offset to the start of a [mini]batch
x = T.matrix() # the data is presented as rasterized images
y = T.ivector() # the labels are presented as 1D vector of
# [int] labels
# construct the logistic regression class
classifier = LogisticRegression(input=x, n_in=28 * 28, n_out=10)
# the cost we minimize during training is the negative log likelihood of
# the model in symbolic format
cost = classifier.negative_log_likelihood(y).mean()
# compile a theano function that computes the mistakes that are made by
# the model on a minibatch
test_model = theano.function(
[minibatch_offset],
classifier.errors(y),
givens={
x: test_set_x[minibatch_offset:minibatch_offset + batch_size],
y: test_set_y[minibatch_offset:minibatch_offset + batch_size]
},
name="test"
)
validate_model = theano.function(
[minibatch_offset],
classifier.errors(y),
givens={
x: valid_set_x[minibatch_offset: minibatch_offset + batch_size],
y: valid_set_y[minibatch_offset: minibatch_offset + batch_size]
},
name="validate"
)
# compile a theano function that returns the cost of a minibatch
batch_cost = theano.function(
[minibatch_offset],
cost,
givens={
x: train_set_x[minibatch_offset: minibatch_offset + batch_size],
y: train_set_y[minibatch_offset: minibatch_offset + batch_size]
},
name="batch_cost"
)
# compile a theano function that returns the gradient of the minibatch
# with respect to theta
batch_grad = theano.function(
[minibatch_offset],
T.grad(cost, classifier.theta),
givens={
x: train_set_x[minibatch_offset: minibatch_offset + batch_size],
y: train_set_y[minibatch_offset: minibatch_offset + batch_size]
},
name="batch_grad"
)
# creates a function that computes the average cost on the training set
def train_fn(theta_value):
classifier.theta.set_value(theta_value, borrow=True)
train_losses = [batch_cost(i * batch_size)
for i in range(n_train_batches)]
return numpy.mean(train_losses)
# creates a function that computes the average gradient of cost with
# respect to theta
def train_fn_grad(theta_value):
classifier.theta.set_value(theta_value, borrow=True)
grad = batch_grad(0)
for i in range(1, n_train_batches):
grad += batch_grad(i * batch_size)
return grad / n_train_batches
validation_scores = [numpy.inf, 0]
# creates the validation function
def callback(theta_value):
classifier.theta.set_value(theta_value, borrow=True)
#compute the validation loss
validation_losses = [validate_model(i * batch_size)
for i in range(n_valid_batches)]
this_validation_loss = numpy.mean(validation_losses)
print('validation error %f %%' % (this_validation_loss * 100.,))
# check if it is better then best validation score got until now
if this_validation_loss < validation_scores[0]:
# if so, replace the old one, and compute the score on the
# testing dataset
validation_scores[0] = this_validation_loss
test_losses = [test_model(i * batch_size)
for i in range(n_test_batches)]
validation_scores[1] = numpy.mean(test_losses)
###############
# TRAIN MODEL #
###############
# using scipy conjugate gradient optimizer
import scipy.optimize
print ("Optimizing using scipy.optimize.fmin_cg...")
start_time = timeit.default_timer()
best_w_b = scipy.optimize.fmin_cg(
f=train_fn,
x0=numpy.zeros((n_in + 1) * n_out, dtype=x.dtype),
fprime=train_fn_grad,
callback=callback,
disp=0,
maxiter=n_epochs
)
end_time = timeit.default_timer()
print(
(
'Optimization complete with best validation score of %f %%, with '
'test performance %f %%'
)
% (validation_scores[0] * 100., validation_scores[1] * 100.)
)
print >> sys.stderr, ('The code for file ' +
os.path.split(__file__)[1] +
' ran for %.1fs' % ((end_time - start_time)))
if __name__ == '__main__':
cg_optimization_mnist()
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