annotate code_tutoriel/logistic_cg.py @ 209:d982dfa583df

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author fsavard
date Fri, 05 Mar 2010 18:08:34 -0500
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1 """
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2 This tutorial introduces logistic regression using Theano and conjugate
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3 gradient descent.
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4
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5 Logistic regression is a probabilistic, linear classifier. It is parametrized
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6 by a weight matrix :math:`W` and a bias vector :math:`b`. Classification is
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7 done by projecting data points onto a set of hyperplanes, the distance to
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8 which is used to determine a class membership probability.
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9
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10 Mathematically, this can be written as:
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11
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12 .. math::
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13 P(Y=i|x, W,b) &= softmax_i(W x + b) \\
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14 &= \frac {e^{W_i x + b_i}} {\sum_j e^{W_j x + b_j}}
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16
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17 The output of the model or prediction is then done by taking the argmax of
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18 the vector whose i'th element is P(Y=i|x).
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19
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20 .. math::
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21
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22 y_{pred} = argmax_i P(Y=i|x,W,b)
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24
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25 This tutorial presents a stochastic gradient descent optimization method
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26 suitable for large datasets, and a conjugate gradient optimization method
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27 that is suitable for smaller datasets.
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29
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30 References:
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31
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32 - textbooks: "Pattern Recognition and Machine Learning" -
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33 Christopher M. Bishop, section 4.3.2
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34
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35
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36 """
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37 __docformat__ = 'restructedtext en'
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38
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39
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40 import numpy, time, cPickle, gzip
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41
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42 import theano
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43 import theano.tensor as T
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44
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45
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46 class LogisticRegression(object):
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47 """Multi-class Logistic Regression Class
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48
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49 The logistic regression is fully described by a weight matrix :math:`W`
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50 and bias vector :math:`b`. Classification is done by projecting data
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51 points onto a set of hyperplanes, the distance to which is used to
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52 determine a class membership probability.
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53 """
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54
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57
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58 def __init__(self, input, n_in, n_out):
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59 """ Initialize the parameters of the logistic regression
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60
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61 :type input: theano.tensor.TensorType
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62 :param input: symbolic variable that describes the input of the
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63 architecture ( one minibatch)
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64
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65 :type n_in: int
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66 :param n_in: number of input units, the dimension of the space in
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67 which the datapoint lies
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68
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69 :type n_out: int
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70 :param n_out: number of output units, the dimension of the space in
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71 which the target lies
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72
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73 """
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74
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75 # initialize theta = (W,b) with 0s; W gets the shape (n_in, n_out),
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76 # while b is a vector of n_out elements, making theta a vector of
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77 # n_in*n_out + n_out elements
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78 self.theta = theano.shared( value = numpy.zeros(n_in*n_out+n_out, dtype = theano.config.floatX) )
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79 # W is represented by the fisr n_in*n_out elements of theta
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80 self.W = self.theta[0:n_in*n_out].reshape((n_in,n_out))
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81 # b is the rest (last n_out elements)
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82 self.b = self.theta[n_in*n_out:n_in*n_out+n_out]
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83
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84
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85 # compute vector of class-membership probabilities in symbolic form
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86 self.p_y_given_x = T.nnet.softmax(T.dot(input, self.W)+self.b)
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87
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88 # compute prediction as class whose probability is maximal in
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89 # symbolic form
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90 self.y_pred=T.argmax(self.p_y_given_x, axis=1)
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91
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93
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95
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96 def negative_log_likelihood(self, y):
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97 """Return the negative log-likelihood of the prediction of this model
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98 under a given target distribution.
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99
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100 .. math::
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101
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102 \frac{1}{|\mathcal{D}|}\mathcal{L} (\theta=\{W,b\}, \mathcal{D}) =
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103 \frac{1}{|\mathcal{D}|}\sum_{i=0}^{|\mathcal{D}|} \log(P(Y=y^{(i)}|x^{(i)}, W,b)) \\
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104 \ell (\theta=\{W,b\}, \mathcal{D})
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105
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106 :type y: theano.tensor.TensorType
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107 :param y: corresponds to a vector that gives for each example the
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108 correct label
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109 """
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110 return -T.mean(T.log(self.p_y_given_x)[T.arange(y.shape[0]),y])
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111
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112
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113
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114
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115
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116 def errors(self, y):
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117 """Return a float representing the number of errors in the minibatch
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118 over the total number of examples of the minibatch
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119
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120 :type y: theano.tensor.TensorType
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121 :param y: corresponds to a vector that gives for each example
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122 the correct label
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123 """
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124
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125 # check if y has same dimension of y_pred
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126 if y.ndim != self.y_pred.ndim:
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127 raise TypeError('y should have the same shape as self.y_pred',
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128 ('y', target.type, 'y_pred', self.y_pred.type))
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129 # check if y is of the correct datatype
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130 if y.dtype.startswith('int'):
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131 # the T.neq operator returns a vector of 0s and 1s, where 1
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132 # represents a mistake in prediction
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133 return T.mean(T.neq(self.y_pred, y))
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134 else:
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135 raise NotImplementedError()
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136
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137
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138
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139
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140
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141
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142
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143 def cg_optimization_mnist( n_epochs=50, mnist_pkl_gz='mnist.pkl.gz' ):
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144 """Demonstrate conjugate gradient optimization of a log-linear model
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145
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146 This is demonstrated on MNIST.
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147
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148 :type n_epochs: int
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149 :param n_epochs: number of epochs to run the optimizer
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150
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151 :type mnist_pkl_gz: string
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152 :param mnist_pkl_gz: the path of the mnist training file from
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153 http://www.iro.umontreal.ca/~lisa/deep/data/mnist/mnist.pkl.gz
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154
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155 """
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156 #############
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157 # LOAD DATA #
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158 #############
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159 print '... loading data'
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160
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161 # Load the dataset
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162 f = gzip.open(mnist_pkl_gz,'rb')
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163 train_set, valid_set, test_set = cPickle.load(f)
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164 f.close()
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165
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166 def shared_dataset(data_xy):
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167 """ Function that loads the dataset into shared variables
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168
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169 The reason we store our dataset in shared variables is to allow
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170 Theano to copy it into the GPU memory (when code is run on GPU).
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171 Since copying data into the GPU is slow, copying a minibatch everytime
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172 is needed (the default behaviour if the data is not in a shared
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173 variable) would lead to a large decrease in performance.
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174 """
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175 data_x, data_y = data_xy
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176 shared_x = theano.shared(numpy.asarray(data_x, dtype=theano.config.floatX))
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177 shared_y = theano.shared(numpy.asarray(data_y, dtype=theano.config.floatX))
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178 # When storing data on the GPU it has to be stored as floats
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179 # therefore we will store the labels as ``floatX`` as well
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180 # (``shared_y`` does exactly that). But during our computations
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181 # we need them as ints (we use labels as index, and if they are
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182 # floats it doesn't make sense) therefore instead of returning
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183 # ``shared_y`` we will have to cast it to int. This little hack
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184 # lets ous get around this issue
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185 return shared_x, T.cast(shared_y, 'int32')
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186
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187
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188 test_set_x, test_set_y = shared_dataset(test_set)
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189 valid_set_x, valid_set_y = shared_dataset(valid_set)
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190 train_set_x, train_set_y = shared_dataset(train_set)
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191
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192 batch_size = 600 # size of the minibatch
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193
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194 n_train_batches = train_set_x.value.shape[0] / batch_size
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195 n_valid_batches = valid_set_x.value.shape[0] / batch_size
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196 n_test_batches = test_set_x.value.shape[0] / batch_size
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197
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198
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199 ishape = (28,28) # this is the size of MNIST images
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200 n_in = 28*28 # number of input units
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201 n_out = 10 # number of output units
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202
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203
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204 ######################
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205 # BUILD ACTUAL MODEL #
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206 ######################
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207 print '... building the model'
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208
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209 # allocate symbolic variables for the data
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210 minibatch_offset = T.lscalar() # offset to the start of a [mini]batch
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211 x = T.matrix() # the data is presented as rasterized images
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212 y = T.ivector() # the labels are presented as 1D vector of
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213 # [int] labels
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214
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215
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216 # construct the logistic regression class
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217 classifier = LogisticRegression( input=x, n_in=28*28, n_out=10)
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218
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219 # the cost we minimize during training is the negative log likelihood of
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220 # the model in symbolic format
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221 cost = classifier.negative_log_likelihood(y).mean()
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222
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223 # compile a theano function that computes the mistakes that are made by
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224 # the model on a minibatch
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225 test_model = theano.function([minibatch_offset], classifier.errors(y),
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226 givens={
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227 x:test_set_x[minibatch_offset:minibatch_offset+batch_size],
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228 y:test_set_y[minibatch_offset:minibatch_offset+batch_size]})
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229
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230 validate_model = theano.function([minibatch_offset],classifier.errors(y),
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231 givens={
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232 x:valid_set_x[minibatch_offset:minibatch_offset+batch_size],
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233 y:valid_set_y[minibatch_offset:minibatch_offset+batch_size]})
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234
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235 # compile a thenao function that returns the cost of a minibatch
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236 batch_cost = theano.function([minibatch_offset], cost,
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237 givens= {
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238 x : train_set_x[minibatch_offset:minibatch_offset+batch_size],
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239 y : train_set_y[minibatch_offset:minibatch_offset+batch_size]})
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240
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241
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242
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243 # compile a theano function that returns the gradient of the minibatch
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244 # with respect to theta
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245 batch_grad = theano.function([minibatch_offset], T.grad(cost,classifier.theta),
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246 givens= {
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247 x : train_set_x[minibatch_offset:minibatch_offset+batch_size],
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248 y : train_set_y[minibatch_offset:minibatch_offset+batch_size]})
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249
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250
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251 # creates a function that computes the average cost on the training set
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252 def train_fn(theta_value):
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253 classifier.theta.value = theta_value
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254 train_losses = [batch_cost(i*batch_size) for i in xrange(n_train_batches)]
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255 return numpy.mean(train_losses)
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256
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257 # creates a function that computes the average gradient of cost with
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258 # respect to theta
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259 def train_fn_grad(theta_value):
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260 classifier.theta.value = theta_value
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261 grad = batch_grad(0)
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262 for i in xrange(1,n_train_batches):
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263 grad += batch_grad(i*batch_size)
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264 return grad/n_train_batches
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265
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266
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267 validation_scores = [float('inf'), 0]
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268
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269 # creates the validation function
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270 def callback(theta_value):
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271 classifier.theta.value = theta_value
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272 #compute the validation loss
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273 validation_losses = [validate_model(i*batch_size) for i in xrange(n_valid_batches)]
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274 this_validation_loss = numpy.mean(validation_losses)
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275 print('validation error %f %%' % (this_validation_loss*100.,))
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276
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277 # check if it is better then best validation score got until now
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278 if this_validation_loss < validation_scores[0]:
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279 # if so, replace the old one, and compute the score on the
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280 # testing dataset
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281 validation_scores[0] = this_validation_loss
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282 test_loses = [test_model(i*batch_size) for i in xrange(n_test_batches)]
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283 validation_scores[1] = numpy.mean(test_loses)
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284
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285 ###############
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286 # TRAIN MODEL #
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287 ###############
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288
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289 # using scipy conjugate gradient optimizer
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290 import scipy.optimize
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291 print ("Optimizing using scipy.optimize.fmin_cg...")
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292 start_time = time.clock()
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293 best_w_b = scipy.optimize.fmin_cg(
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294 f = train_fn,
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295 x0 = numpy.zeros((n_in+1)*n_out, dtype=x.dtype),
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296 fprime = train_fn_grad,
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297 callback = callback,
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298 disp = 0,
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299 maxiter = n_epochs)
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300 end_time = time.clock()
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301 print(('Optimization complete with best validation score of %f %%, with '
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302 'test performance %f %%') %
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303 (validation_scores[0]*100., validation_scores[1]*100.))
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304
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305 print ('The code ran for %f minutes' % ((end_time-start_time)/60.))
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306
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307
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308 if __name__ == '__main__':
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309 cg_optimization_mnist()
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310