ift6266: code_tutoriel/logistic

comparison code_tutoriel/logistic_cg.py @ 0:fda5f787baa6

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author	Dumitru Erhan <dumitru.erhan@gmail.com>
date	Thu, 21 Jan 2010 11:26:43 -0500
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+:fda5f787baa6
+"""
+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 stochastic gradient descent optimization method
+suitable for large datasets, and 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 numpy, cPickle, gzip
+import time
+import theano
+import theano.tensor as T
+import theano.tensor.nnet
+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
+:param input: symbolic variable that describes the input of the
+architecture ( one minibatch)
+:param n_in: number of input units, the dimension of the space in
+which the datapoint lies
+: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) )
+# 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)
+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})
+: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
+"""
+# 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', target.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_iter=50 ):
+"""Demonstrate conjugate gradient optimization of a log-linear model
+This is demonstrated on MNIST.
+:param n_iter: number of iterations ot run the optimizer
+"""
+# Load the dataset
+f = gzip.open('mnist.pkl.gz','rb')
+train_set, valid_set, test_set = cPickle.load(f)
+f.close()
+# make minibatches of size 20
+batch_size = 20    # sized of the minibatch
+# Dealing with the training set
+# get the list of training images (x) and their labels (y)
+(train_set_x, train_set_y) = train_set
+# initialize the list of training minibatches with empty list
+train_batches = []
+for i in xrange(0, len(train_set_x), batch_size):
+# add to the list of minibatches the minibatch starting at
+# position i, ending at position i+batch_size
+# a minibatch is a pair ; the first element of the pair is a list
+# of datapoints, the second element is the list of corresponding
+# labels
+train_batches = train_batches + \
+[(train_set_x[i:i+batch_size], train_set_y[i:i+batch_size])]
+# Dealing with the validation set
+(valid_set_x, valid_set_y) = valid_set
+# initialize the list of validation minibatches
+valid_batches = []
+for i in xrange(0, len(valid_set_x), batch_size):
+valid_batches = valid_batches + \
+[(valid_set_x[i:i+batch_size], valid_set_y[i:i+batch_size])]
+# Dealing with the testing set
+(test_set_x, test_set_y) = test_set
+# initialize the list of testing minibatches
+test_batches = []
+for i in xrange(0, len(test_set_x), batch_size):
+test_batches = test_batches + \
+[(test_set_x[i:i+batch_size], test_set_y[i:i+batch_size])]
+ishape     = (28,28) # this is the size of MNIST images
+n_in       = 28*28   # number of input units
+n_out      = 10      # number of output units
+# allocate symbolic variables for the data
+x = T.fmatrix()  # the data is presented as rasterized images
+y = T.lvector()  # the labels are presented as 1D vector of
+# [long int] labels
+# construct the logistic regression class
+classifier = LogisticRegression( \
+input=x.reshape((batch_size,28*28)), 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([x,y], classifier.errors(y))
+# compile a theano function that returns the gradient of the minibatch
+# with respect to theta
+batch_grad = theano.function([x, y], T.grad(cost, classifier.theta))
+#  compile a thenao function that returns the cost of a minibatch
+batch_cost = theano.function([x, y], cost)
+# creates a function that computes the average cost on the training set
+def train_fn(theta_value):
+classifier.theta.value = theta_value
+cost = 0.
+for x,y in train_batches :
+cost += batch_cost(x,y)
+return cost / len(train_batches)
+# creates a function that computes the average gradient of cost with
+# respect to theta
+def train_fn_grad(theta_value):
+classifier.theta.value = theta_value
+grad = numpy.zeros(n_in * n_out + n_out)
+for x,y in train_batches:
+grad += batch_grad(x,y)
+return grad/ len(train_batches)
+validation_scores = [float('inf'), 0]
+# creates the validation function
+def callback(theta_value):
+classifier.theta.value = theta_value
+#compute the validation loss
+this_validation_loss = 0.
+for x,y in valid_batches:
+this_validation_loss += test_model(x,y)
+this_validation_loss /= len(valid_batches)
+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_score = 0.
+for x,y in test_batches:
+test_score += test_model(x,y)
+validation_scores[1] = test_score / len(test_batches)
+# using scipy conjugate gradient optimizer
+import scipy.optimize
+print ("Optimizing using scipy.optimize.fmin_cg...")
+start_time = time.clock()
+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_iter)
+end_time = time.clock()
+print(('Optimization complete with best validation score of %f %%, with '
+'test performance %f %%') %
+(validation_scores[0]*100., validation_scores[1]*100.))
+print ('The code ran for %f minutes' % ((end_time-start_time)/60.))
+if __name__ == '__main__':
+cg_optimization_mnist()

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comparison code_tutoriel/logistic_cg.py @ 0:fda5f787baa6