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annotate pylearn/algorithms/mcRBM.py @ 1267:075c193afd1b

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refactoring mcRBM
author James Bergstra <bergstrj@iro.umontreal.ca>
date Fri, 03 Sep 2010 12:35:10 -0400
parents d4a14c6c36e0
children d38cb039c662
rev   line source
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1 """
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2 This file implements the Mean & Covariance RBM discussed in
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3
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4 Ranzato, M. and Hinton, G. E. (2010)
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5 Modeling pixel means and covariances using factored third-order Boltzmann machines.
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6 IEEE Conference on Computer Vision and Pattern Recognition.
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7
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8 and performs one of the experiments on CIFAR-10 discussed in that paper. There are some minor
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9 discrepancies between the paper and the accompanying code (train_mcRBM.py), and the
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10 accompanying code has been taken to be correct in those cases because I couldn't get things to
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11 work otherwise.
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12
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13
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14 Math
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15 ====
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16
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17 Energy of "covariance RBM"
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18
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19 E = -0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i C_{if} v_i )^2
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20 = -0.5 \sum_f (\sum_k P_{fk} h_k) ( \sum_i C_{if} v_i )^2
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21 "vector element f" "vector element f"
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22
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23 In some parts of the paper, the P matrix is chosen to be a diagonal matrix with non-positive
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24 diagonal entries, so it is helpful to see this as a simpler equation:
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25
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26 E = \sum_f h_f ( \sum_i C_{if} v_i )^2
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27
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28
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29
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30 Version in paper
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31 ----------------
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32
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33 Full Energy of the Mean and Covariance RBM, with
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34 :math:`h_k = h_k^{(c)}`,
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35 :math:`g_j = h_j^{(m)}`,
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36 :math:`b_k = b_k^{(c)}`,
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37 :math:`c_j = b_j^{(m)}`,
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38 :math:`U_{if} = C_{if}`,
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39
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40 E (v, h, g) =
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41 - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i (U_{if} v_i) / |U_{.f}|*|v| )^2
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42 - \sum_k b_k h_k
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43 + 0.5 \sum_i v_i^2
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44 - \sum_j \sum_i W_{ij} g_j v_i
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45 - \sum_j c_j g_j
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46
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47 For the energy function to correspond to a probability distribution, P must be non-positive. P
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48 is initialized to be a diagonal, and in our experience it can be left as such because even in
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49 the paper it has a very low learning rate, and is only allowed to be updated after the filters
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50 in U are learned (in effect).
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51
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52 Version in published train_mcRBM code
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53 -------------------------------------
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54
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55 The train_mcRBM file implements learning in a similar but technically different Energy function:
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56
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57 E (v, h, g) =
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58 - 0.5 \sum_f \sum_k P_{fk} h_k (\sum_i U_{if} v_i / sqrt(\sum_i v_i^2/I + 0.5))^2
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59 - \sum_k b_k h_k
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60 + 0.5 \sum_i v_i^2
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61 - \sum_j \sum_i W_{ij} g_j v_i
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62 - \sum_j c_j g_j
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63
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64 There are two differences with respect to the paper:
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65
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66 - 'v' is not normalized by its length, but rather it is normalized to have length close to
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67 the square root of the number of its components. The variable called 'small' that
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68 "avoids division by zero" is orders larger than machine precision, and is on the order of
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69 the normalized sum-of-squares, so I've included it in the Energy function.
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70
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71 - 'U' is also not normalized by its length. U is initialized to have columns that are
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72 shorter than unit-length (approximately 0.2 with the 105 principle components in the
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73 train_mcRBM data). During training, the columns of U are constrained manually to have
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74 equal lengths (see the use of normVF), but Euclidean norm is allowed to change. During
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75 learning it quickly converges towards 1 and then exceeds 1. It does not seem like this
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76 column-wise normalization of U is justified by maximum-likelihood, I have no intuition
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77 for why it is used.
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78
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79
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80 Version in this code
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81 --------------------
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82
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83 This file implements the same algorithm as the train_mcRBM code, except that the P matrix is
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84 omitted for clarity, and replaced analytically with a negative identity matrix.
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85
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86 E (v, h, g) =
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87 + 0.5 \sum_k h_k (\sum_i U_{ik} v_i / sqrt(\sum_i v_i^2/I + 0.5))^2
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88 - \sum_k b_k h_k
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89 + 0.5 \sum_i v_i^2
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90 - \sum_j \sum_i W_{ij} g_j v_i
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91 - \sum_j c_j g_j
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92
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93
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94
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95 Conventions in this file
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96 ========================
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97
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98 This file contains some global functions, as well as a class (MeanCovRBM) that makes using them a little
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99 more convenient.
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100
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101
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102 Global functions like `free_energy` work on an mcRBM as parametrized in a particular way.
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103 Suppose we have
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104 I input dimensions,
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105 F squared filters,
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106 J mean variables, and
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107 K covariance variables.
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108 The mcRBM is parametrized by 5 variables:
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109
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110 - `U`, a matrix whose rows are visible covariance directions (I x F)
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111 - `W`, a matrix whose rows are visible mean directions (I x J)
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112 - `b`, a vector of hidden covariance biases (K)
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113 - `c`, a vector of hidden mean biases (J)
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114
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115 Matrices are generally layed out and accessed according to a C-order convention.
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116
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117 """
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118
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119 #
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120 # WORKING NOTES
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121 # THIS DERIVATION IS BASED ON THE ** PAPER ** ENERGY FUNCTION
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122 # NOT THE ENERGY FUNCTION IN THE CODE!!!
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123 #
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124 # Free energy is the marginal energy of visible units
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125 # Recall:
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126 # Q(x) = exp(-E(x))/Z ==> -log(Q(x)) - log(Z) = E(x)
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127 #
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128 #
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129 # E (v, h, g) =
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130 # - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / |U_{*f}|^2 |v|^2
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131 # - \sum_k b_k h_k
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132 # + 0.5 \sum_i v_i^2
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133 # - \sum_j \sum_i W_{ij} g_j v_i
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134 # - \sum_j c_j g_j
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135 # - \sum_i a_i v_i
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136 #
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137 #
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138 # Derivation, in which partition functions are ignored.
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139 #
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140 # E(v) = -\log(Q(v))
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141 # = -\log( \sum_{h,g} Q(v,h,g))
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142 # = -\log( \sum_{h,g} exp(-E(v,h,g)))
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143 # = -\log( \sum_{h,g} exp(-
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144 # - 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}| * |v|)
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145 # - \sum_k b_k h_k
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146 # + 0.5 \sum_i v_i^2
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147 # - \sum_j \sum_i W_{ij} g_j v_i
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148 # - \sum_j c_j g_j
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149 # - \sum_i a_i v_i ))
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150 #
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151 # Get rid of double negs in exp
967
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152 # = -\log( \sum_{h} exp(
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153 # + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}| * |v|)
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154 # + \sum_k b_k h_k
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155 # - 0.5 \sum_i v_i^2
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156 # ) * \sum_{g} exp(
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157 # + \sum_j \sum_i W_{ij} g_j v_i
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158 # + \sum_j c_j g_j))
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159 # - \sum_i a_i v_i
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160 #
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161 # Break up log
967
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162 # = -\log( \sum_{h} exp(
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163 # + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}|*|v|)
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164 # + \sum_k b_k h_k
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165 # ))
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166 # -\log( \sum_{g} exp(
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167 # + \sum_j \sum_i W_{ij} g_j v_i
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168 # + \sum_j c_j g_j )))
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169 # + 0.5 \sum_i v_i^2
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170 # - \sum_i a_i v_i
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171 #
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172 # Use domain h is binary to turn log(sum(exp(sum...))) into sum(log(..
967
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173 # = -\log(\sum_{h} exp(
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174 # + 0.5 \sum_f \sum_k P_{fk} h_k ( \sum_i U_{if} v_i )^2 / (|U_{*f}|* |v|)
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175 # + \sum_k b_k h_k
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176 # ))
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177 # - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j ))
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178 # + 0.5 \sum_i v_i^2
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179 # - \sum_i a_i v_i
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180 #
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181 # = - \sum_{k} \log(1 + exp(b_k + 0.5 \sum_f P_{fk}( \sum_i U_{if} v_i )^2 / (|U_{*f}|*|v|)))
967
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182 # - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j ))
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183 # + 0.5 \sum_i v_i^2
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184 # - \sum_i a_i v_i
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185 #
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186 # For negative-one-diagonal P this gives:
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187 #
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188 # = - \sum_{k} \log(1 + exp(b_k - 0.5 \sum_i (U_{ik} v_i )^2 / (|U_{*k}|*|v|)))
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189 # - \sum_{j} \log(1 + exp(\sum_i W_{ij} v_i + c_j ))
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190 # + 0.5 \sum_i v_i^2
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191 # - \sum_i a_i v_i
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192
1000
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193 import sys, os, logging
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194 import numpy as np
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195 import numpy
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196
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197 import theano
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198 from theano import function, shared, dot
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199 from theano import tensor as TT
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200 floatX = theano.config.floatX
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201
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202 import pylearn
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203 #TODO: clean up the HMC_sampler code
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204 #TODO: think of naming convention for acronyms + suffix?
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205 from pylearn.sampling.hmc import HMC_sampler
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206 from pylearn.io import image_tiling
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207 from pylearn.gd.sgd import sgd_updates
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208 import pylearn.dataset_ops.image_patches
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209
1000
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210 ###########################################
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211 #
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212 # Candidates for factoring
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213 #
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214 ###########################################
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215
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216 #TODO: Document, move to pylearn's math lib
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217 def l1(X):
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218 return abs(X).sum()
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219
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220 #TODO: Document, move to pylearn's math lib
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221 def l2(X):
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222 return TT.sqrt((X**2).sum())
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223
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224 #TODO: Document, move to pylearn's math lib
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225 def contrastive_cost(free_energy_fn, pos_v, neg_v):
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226 return (free_energy_fn(pos_v) - free_energy_fn(neg_v)).sum()
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227
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228 #TODO: Typical use of contrastive_cost is to later use tensor.grad, but in that case we want to
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229 # block gradient going through neg_v
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230 def contrastive_grad(free_energy_fn, pos_v, neg_v, params, other_cost=0):
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231 """
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232 :param pos_v: positive-phase sample of visible units
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233 :param neg_v: negative-phase sample of visible units
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234 """
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235 #block the grad through neg_v
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236 cost=contrastive_cost(free_energy_fn, pos_v, neg_v)
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237 if other_cost:
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238 cost = cost + other_cost
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239 return theano.tensor.grad(cost,
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240 wrt=params,
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241 consider_constant=[neg_v])
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242
1000
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243 ###########################################
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244 #
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245 # Expressions that are mcRBM-specific
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246 #
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247 ###########################################
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248
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249 class mcRBM(object):
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250 """Light-weight class that provides the math related to inference
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251
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252 Attributes:
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253
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254 - U - the covariance filters (theano shared variable)
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255 - W - the mean filters (theano shared variable)
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256 - a - the visible bias (theano shared variable)
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257 - b - the covariance bias (theano shared variable)
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258 - c - the mean bias (theano shared variable)
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259 """
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260 def __init__(self, U, W, a, b, c):
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261 self.U = U
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262 self.W = W
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263 self.a = a
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264 self.b = b
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265 self.c = c
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266
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267 def hidden_cov_units_preactivation_given_v(self, v, small=0.5):
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268 """Return argument to the sigmoid that would give mean of covariance hid units
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269
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270 See the math at the top of this file for what 'adjusted' means.
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271
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272 return b - 0.5 * dot(adjusted(v), U)**2
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273 """
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274 unit_v = v / (TT.sqrt(TT.mean(v**2, axis=1)+small)).dimshuffle(0,'x') # adjust row norm
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275 return self.b - 0.5 * dot(unit_v, self.U)**2
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276
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277 def free_energy_terms_given_v(self, v):
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278 """Returns theano expression for the terms that are added to form the free energy of
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279 visible vector `v` in an mcRBM.
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280
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281 1. Free energy related to covariance hiddens
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282 2. Free energy related to mean hiddens
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283 3. Free energy related to L2-Norm of `v`
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284 4. Free energy related to projection of `v` onto biases `a`
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285 """
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286 t0 = -TT.sum(TT.nnet.softplus(self.hidden_cov_units_preactivation_given_v(v)),axis=1)
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287 t1 = -TT.sum(TT.nnet.softplus(self.c + dot(v,self.W)), axis=1)
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288 t2 = 0.5 * TT.sum(v**2, axis=1)
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289 t3 = -TT.dot(v, self.a)
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290 return [t0, t1, t2, t3]
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291
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292 def free_energy_given_v(self, v):
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293 """Returns theano expression for free energy of visible vector `v` in an mcRBM
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294 """
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295 return TT.add(*self.free_energy_terms_given_v(v))
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296
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297 def expected_h_g_given_v(self, v):
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298 """Returns tuple (`h`, `g`) of theano expression conditional expectations in an mcRBM.
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299
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300 `h` is the conditional on the covariance units.
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301 `g` is the conditional on the mean units.
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302
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303 """
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304 h = TT.nnet.sigmoid(self.hidden_cov_units_preactivation_given_v(v))
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305 g = nnet.sigmoid(self.c + dot(v,self.W))
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306 return (h, g)
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307
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308 def n_visible_units(self):
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309 """Return the number of visible units of this RBM
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310
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311 For an RBM made from shared variables, this will return an integer,
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312 for a purely symbolic RBM this will return a theano expression.
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313
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314 """
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315 try:
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316 return self.W.value.shape[0]
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317 except AttributeError:
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318 return self.W.shape[0]
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319
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320 def sampler(self, n_particles, n_visible=None, rng=7823748):
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321 """Return an `HMC_sampler` that will draw samples from the distribution over visible
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322 units specified by this RBM.
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323
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324 :param n_particles: this many parallel chains will be simulated.
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325 :param rng: seed or numpy RandomState object to initialize particles, and to drive the simulation.
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326 """
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327 if not hasattr(rng, 'randn'):
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328 rng = np.random.RandomState(rng)
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329 if n_visible is None:
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330 n_visible = self.n_visible_units()
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331 rval = HMC_sampler(
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332 positions = [shared(
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333 rng.randn(
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334 n_particles,
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335 n_visible).astype(floatX),
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336 name='particles')],
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337 energy_fn=self.free_energy_given_v,
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338 seed=int(rng.randint(2**30)))
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339 return rval
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340
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341 @classmethod
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342 def alloc(cls, n_I, n_K, n_J, rng = 8923402190):
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343 """
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344 Return a MeanCovRBM instance with randomly-initialized parameters.
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345
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346 :param n_I: input dimensionality
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347 :param n_K: number of covariance hidden units
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348 :param n_J: number of mean filters (linear)
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349 :param rng: seed or numpy RandomState object to initialize params
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350 """
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351 if not hasattr(rng, 'randn'):
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352 rng = np.random.RandomState(rng)
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353
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354 def shrd(X,name):
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355 return shared(X.astype(floatX), name=name)
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356
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357 # initialization taken from train_mcRBM.py
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358 rval = cls(
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359 U = shrd(0.02 * rng.randn(n_I, n_K),'U'),
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360 W = shrd(0.05 * rng.randn(n_I, n_J),'W'),
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361 a = shrd(np.ones(n_I)*(0),'a'),
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362 b = shrd(np.ones(n_K)*2,'b'),
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363 c = shrd(np.ones(n_J)*(-2),'c'))
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364
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365 rval.params = [rval.U, rval.W, rval.a, rval.b, rval.c]
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366 return rval
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367
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368 class mcRBMTrainer(object):
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369 """
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370
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371 Attributes:
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372 - rbm
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373 - sampler
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374 - normVF
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375 - learn_rate
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376 - learn_rate_multipliers
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377
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378 """
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379 def __init__(self, **kwargs):
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380 self.__dict__.update(kwargs)
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381
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382 def normalize_U(self, new_U):
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383 #TODO: write the docstring
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384 U_norms = TT.sqrt((new_U**2).sum(axis=0))
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385 new_normVF = .95 * self.normVF + .05 * TT.mean(U_norms)
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386 return new_U * this_normVF / U_norms), new_normVF
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387
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388 def contrastive_grads(self, visible_batch, params=None):
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389 if params is not None:
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390 params = self.rbm.params
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391 return contrastive_grad(
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392 free_energy_fn=rbm.free_energy_given_v,
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393 pos_v=visible_batch,
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394 neg_v=self.sampler.positions,
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395 params=params,
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396 other_cost=(l1(self.rbm.U)+l1(self.rbm.W)) * self.l1_penalty)
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397
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398
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399 def cd_updates(self, visible_batch, params=None, rng=89234):
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400 if params is not None:
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401 params = self.rbm.params
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402
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403 grads = self.contrastive_grads(visible_batch, params)
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404
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405 # contrastive divergence updates
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406 # TODO: sgd_updates is a particular optization algo (others are possible)
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407 # parametrize so that algo is plugin
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408 # the normalization normVF might be sgd-specific though...
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409
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410 # TODO: when sgd has an annealing schedule, this should
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411 # go through that mechanism.
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412
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413 # TODO: parametrize these constants (e.g. 2000)
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414
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415 ups[self.iter] = self.iter + 1
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416 lr = TT.clip(
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417 self.learn_rate * 2000 / (self.iter+1),
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418 0.0, #min
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419 self.learn_rate) #max
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420
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421 ups = sgd_updates(
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422 params,
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423 grads,
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424 stepsizes=[a*lr for a in learn_rate_multipliers])
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425
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426 # sampler updates
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427 ups.update(dict(self.sampler.updates()))
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428
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429 # add trainer updates (replace CD update of U)
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430 ups[self.rbm.U], ups[self.normVF] = self.normalize_U(ups[U])
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431
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432 return ups
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433
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434 # TODO: accept a GD algo as an argument?
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435 @classmethod
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436 def alloc(cls, rbm, visible_batch, batchsize, initial_lr=0.075, rng=234,
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437 l1_penalty=0,
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438 learn_rate_multipliers=[2, .2, .02, .1, .02]):
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439 # allocates shared var for negative phase particles
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440
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441 return cls(
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442 rbm=rbm,
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443 sampler=rbm.sampler(batchsize, rng=rng),
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444 normVF=shared(1.0, 'normVF'),
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445 learn_rate=shared(initial_lr/batchsize, 'learn_rate'),
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446 iter=shared(0, 'iter'),
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447 l1_penalty=l1_penalty,
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448 learn_rate_multipliers=learn_rate_multipliers)
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449
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450
1000
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451 if __name__ == '__main__':
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452 import pylearn.algorithms.tests.test_mcRBM
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453 pylearn.algorithms.tests.test_mcRBM.test_reproduce_ranzato_hinton_2010(as_unittest=True)