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annotate linear_regression.py @ 363:9e84e8a20a75

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author Joseph Turian <turian@gmail.com>
date Thu, 03 Jul 2008 17:52:11 -0400
parents f6505ec32dc3
children c9a89be5cb0a
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1 """
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2 Implementation of linear regression, with or without L2 regularization.
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3 This is one of the simplest example of L{learner}, and illustrates
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4 the use of theano.
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5 """
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7 from learner import *
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8 from theano import tensor as t
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9 from theano.scalar import as_scalar
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10
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11 class LinearRegression(MinibatchUpdatesTLearner):
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12 """
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13 Implement linear regression, with or without L2 regularization
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14 (the former is called Ridge Regression and the latter Ordinary Least Squares).
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15
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16 The predictor parameters are obtained analytically from the training set.
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17 Training can proceed sequentially (with multiple calls to update with
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18 different disjoint subsets of the training sets). After each call to
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19 update the predictor is ready to be used (and optimized for the union
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20 of all the training sets passed to update since construction or since
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21 the last call to forget).
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22
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23 For each (input[t],output[t]) pair in a minibatch,::
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24
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25 output_t = b + W * input_t
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26
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27 where b and W are obtained by minimizing::
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28
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29 L2_regularizer sum_{ij} W_{ij}^2 + sum_t ||output_t - target_t||^2
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30
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31 Let X be the whole training set inputs matrix (one input example per row),
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32 with the first column full of 1's, and Let Y the whole training set
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33 targets matrix (one example's target vector per row).
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34 Let theta = the matrix with b in its first column and W in the others,
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35 then each theta[:,i] is the solution of the linear system::
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36
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37 XtX * theta[:,i] = XtY[:,i]
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38
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39 where XtX is a (n_inputs+1)x(n_inputs+1) matrix containing X'*X
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40 plus L2_regularizer on the diagonal except at (0,0),
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41 and XtY is a (n_inputs+1)*n_outputs matrix containing X'*Y.
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42
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43 The fields and attributes expected and produced by use and update are the following:
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44
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45 - Input and output fields (example-wise quantities):
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46
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47 - 'input' (always expected by use and update as an input_dataset field)
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48 - 'target' (optionally expected by use and update as an input_dataset field)
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49 - 'output' (optionally produced by use as an output dataset field)
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50 - 'squared_error' (optionally produced by use as an output dataset field, needs 'target') = example-wise squared error
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51
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52 - optional attributes (optionally expected as input_dataset attributes)
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53 (warning, this may be dangerous, the 'use' method will use those provided in the
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54 input_dataset rather than those learned during 'update'; currently no support
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55 for providing these to update):
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57 - 'L2_regularizer'
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58 - 'b'
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59 - 'W'
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60 - 'parameters' = [b, W]
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61 - 'regularization_term'
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62 - 'XtX'
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63 - 'XtY'
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65 """
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66
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67 def attributeNames(self):
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68 return ["L2_regularizer","parameters","b","W","regularization_term","XtX","XtY"]
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69
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70 def useInputAttributes(self):
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71 return ["b","W"]
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72
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73 def useOutputAttributes(self):
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74 return []
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76 def updateInputAttributes(self):
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77 return ["L2_regularizer","XtX","XtY"]
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79 def updateMinibatchInputFields(self):
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80 return ["input","target"]
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81
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82 def updateMinibatchInputAttributes(self):
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83 return ["XtX","XtY"]
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84
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85 def updateMinibatchOutputAttributes(self):
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86 return ["new_XtX","new_XtY"]
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87
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88 def updateEndInputAttributes(self):
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89 return ["theta","XtX","XtY"]
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91 def updateEndOutputAttributes(self):
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92 return ["new_theta","b","W","regularization_term"] # CHECK: WILL b AND W CONTAIN OLD OR NEW THETA? @todo i.e. order of computation = ?
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94 def parameterAttributes(self):
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95 return ["b","W"]
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96
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97 def defaultOutputFields(self, input_fields):
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98 output_fields = ["output"]
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99 if "target" in input_fields:
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100 output_fields.append("squared_error")
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101 return output_fields
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102
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103 def __init__(self):
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104 self._input = t.matrix('input') # n_examples x n_inputs
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105 self._target = t.matrix('target') # n_examples x n_outputs
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106 self._L2_regularizer = as_scalar(0.,'L2_regularizer')
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107 self._theta = t.matrix('theta')
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108 self._W = self._theta[:,1:]
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109 self._b = self._theta[:,0]
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110 self._XtX = t.matrix('XtX')
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111 self._XtY = t.matrix('XtY')
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112 self._extended_input = t.prepend_one_to_each_row(self._input)
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113 self._output = t.dot(self._input,self._W.T) + self._b # (n_examples , n_outputs) matrix
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114 self._squared_error = t.sum_within_rows(t.sqr(self._output-self._target)) # (n_examples ) vector
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115 self._regularizer = self._L2_regularizer * t.dot(self._W,self._W)
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116 self._new_XtX = add_inplace(self._XtX,t.dot(self._extended_input.T,self._extended_input))
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117 self._new_XtY = add_inplace(self._XtY,t.dot(self._extended_input.T,self._target))
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118 self._new_theta = t.solve_inplace(self._theta,self._XtX,self._XtY)
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119
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120 MinibatchUpdatesTLearner.__init__(self)
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121
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122 def allocate(self,minibatch):
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123 minibatch_n_inputs = minibatch["input"].shape[1]
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124 minibatch_n_outputs = minibatch["target"].shape[1]
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125 if not self._n_inputs:
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126 self._n_inputs = minibatch_n_inputs
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127 self._n_outputs = minibatch_n_outputs
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128 self.XtX = numpy.zeros((1+self._n_inputs,1+self._n_inputs))
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129 self.XtY = numpy.zeros((1+self._n_inputs,self._n_outputs))
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130 self.theta = numpy.zeros((self._n_outputs,1+self._n_inputs))
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131 self.forget()
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132 elif self._n_inputs!=minibatch_n_inputs or self._n_outputs!=minibatch_n_outputs:
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133 # if the input or target changes dimension on the fly, we resize and forget everything
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134 self.forget()
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135
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136 def forget(self):
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137 if self._n_inputs and self._n_outputs:
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138 self.XtX.resize((1+self.n_inputs,1+self.n_inputs))
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139 self.XtY.resize((1+self.n_inputs,self.n_outputs))
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140 self.XtX.data[:,:]=0
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141 self.XtY.data[:,:]=0
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142 numpy.diag(self.XtX.data)[1:]=self.L2_regularizer
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143