annotate writeup/techreport.tex @ 479:6593e67381a3

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author Xavier Glorot <glorotxa@iro.umontreal.ca>
date Sun, 30 May 2010 18:54:36 -0400
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1 \documentclass[12pt,letterpaper]{article}
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2 \usepackage[utf8]{inputenc}
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3 \usepackage{graphicx}
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4 \usepackage{times}
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5 \usepackage{mlapa}
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6 \usepackage{subfigure}
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7
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8 \begin{document}
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9 \title{Generating and Exploiting Perturbed and Multi-Task Handwritten Training Data for Deep Architectures}
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10 \author{The IFT6266 Gang}
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11 \date{April 2010, Technical Report, Dept. IRO, U. Montreal}
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12
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13 \maketitle
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14
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15 \begin{abstract}
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16 Recent theoretical and empirical work in statistical machine learning has
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17 demonstrated the importance of learning algorithms for deep
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18 architectures, i.e., function classes obtained by composing multiple
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19 non-linear transformations. In the area of handwriting recognition,
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20 deep learning algorithms
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21 had been evaluated on rather small datasets with a few tens of thousands
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22 of examples. Here we propose a powerful generator of variations
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23 of examples for character images based on a pipeline of stochastic
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24 transformations that include not only the usual affine transformations
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25 but also the addition of slant, local elastic deformations, changes
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26 in thickness, background images, color, contrast, occlusion, and
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27 various types of pixel and spatially correlated noise.
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28 We evaluate a deep learning algorithm (Stacked Denoising Autoencoders)
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29 on the task of learning to classify digits and letters transformed
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30 with this pipeline, using the hundreds of millions of generated examples
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31 and testing on the full 62-class NIST test set.
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32 We find that the SDA outperforms its
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33 shallow counterpart, an ordinary Multi-Layer Perceptron,
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34 and that it is better able to take advantage of the additional
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35 generated data, as well as better able to take advantage of
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36 the multi-task setting, i.e.,
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37 training from more classes than those of interest in the end.
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38 In fact, we find that the SDA reaches human performance as
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39 estimated by the Amazon Mechanical Turk on the 62-class NIST test characters.
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40 \end{abstract}
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41
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42 \section{Introduction}
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43
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44 Deep Learning has emerged as a promising new area of research in
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45 statistical machine learning (see~\emcite{Bengio-2009} for a review).
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46 Learning algorithms for deep architectures are centered on the learning
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47 of useful representations of data, which are better suited to the task at hand.
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48 This is in great part inspired by observations of the mammalian visual cortex,
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49 which consists of a chain of processing elements, each of which is associated with a
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50 different representation. In fact,
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51 it was found recently that the features learnt in deep architectures resemble
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52 those observed in the first two of these stages (in areas V1 and V2
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53 of visual cortex)~\cite{HonglakL2008}.
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54 Processing images typically involves transforming the raw pixel data into
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55 new {\bf representations} that can be used for analysis or classification.
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56 For example, a principal component analysis representation linearly projects
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57 the input image into a lower-dimensional feature space.
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58 Why learn a representation? Current practice in the computer vision
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59 literature converts the raw pixels into a hand-crafted representation
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60 (e.g.\ SIFT features~\cite{Lowe04}), but deep learning algorithms
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61 tend to discover similar features in their first few
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62 levels~\cite{HonglakL2008,ranzato-08,Koray-08,VincentPLarochelleH2008-very-small}.
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63 Learning increases the
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64 ease and practicality of developing representations that are at once
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65 tailored to specific tasks, yet are able to borrow statistical strength
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66 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
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67 feature representation can lead to higher-level (more abstract, more
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68 general) features that are more robust to unanticipated sources of
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69 variance extant in real data.
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70
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71 Whereas a deep architecture can in principle be more powerful than a
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72 shallow one in terms of representation, depth appears to render the
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73 training problem more difficult in terms of optimization and local minima.
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74 It is also only recently that successful algorithms were proposed to
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75 overcome some of these difficulties. All are based on unsupervised
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76 learning, often in an greedy layer-wise ``unsupervised pre-training''
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77 stage~\cite{Bengio-2009}. One of these layer initialization techniques,
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78 applied here, is the Denoising
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79 Auto-Encoder~(DEA)~\cite{VincentPLarochelleH2008-very-small}, which
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80 performed similarly or better than previously proposed Restricted Boltzmann
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81 Machines in terms of unsupervised extraction of a hierarchy of features
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82 useful for classification. The principle is that each layer starting from
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83 the bottom is trained to encode their input (the output of the previous
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84 layer) and try to reconstruct it from a corrupted version of it. After this
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85 unsupervised initialization, the stack of denoising auto-encoders can be
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86 converted into a deep supervised feedforward neural network and trained by
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87 stochastic gradient descent.
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88
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89
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90 \section{Perturbation and Transformation of Character Images}
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91
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92 This section describes the different transformations we used to generate data, in their order.
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93 The code for these transformations (mostly python) is available at
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94 {\tt http://anonymous.url.net}. All the modules in the pipeline share
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95 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
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96 amount of deformation or noise introduced.
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97
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98 We can differentiate two important parts in the pipeline. The first one,
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99 from slant to pinch, performs transformations of the character. The second
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100 part, from blur to contrast, adds noise to the image.
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101
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102 \subsection{Slant}
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103
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104 In order to mimic a slant effect, we simply shift each row of the image
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105 proportionnaly to its height: $shift = round(slant \times height)$. We
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106 round the shift in order to have a discret displacement. We do not use a
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107 filter to smooth the result in order to save computing time and also
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108 because latter transformations have similar effects.
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109
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110 The $slant$ coefficient can be negative or positive with equal probability
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111 and its value is randomly sampled according to the complexity level. In
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112 our case we take uniformly a number in the range $[0,complexity]$, so the
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113 maximum displacement for the lowest or highest pixel line is of
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114 $round(complexity \times 32)$.
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115
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116
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117 \subsection{Thickness}
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118
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119 To change the thickness of the characters we used morpholigical operators:
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120 dilation and erosion~\cite{Haralick87,Serra82}.
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121
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122 The basic idea of such transform is, for each pixel, to multiply in the
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123 element-wise manner its neighbourhood with a matrix called the structuring
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124 element. Then for dilation we remplace the pixel value by the maximum of
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125 the result, or the minimum for erosion. This will dilate or erode objects
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126 in the image and the strength of the transform only depends on the
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127 structuring element.
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128
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129 We used ten different structural elements with increasing dimensions (the
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130 biggest is $5\times5$). for each image, we radomly sample the operator
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131 type (dilation or erosion) with equal probability and one structural
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132 element from a subset of the $n$ smallest structuring elements where $n$ is
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133 $round(10 \times complexity)$ for dilation and $round(6 \times complexity)$
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134 for erosion. A neutral element is always present in the set, if it is
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135 chosen the transformation is not applied. Erosion allows only the six
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136 smallest structural elements because when the character is too thin it may
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137 erase it completly.
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138
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139 \subsection{Affine Transformations}
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140
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141 We generate an affine transform matrix according to the complexity level,
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142 then we apply it directly to the image. The matrix is of size $2 \times
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143 3$, so we can represent it by six parameters $(a,b,c,d,e,f)$. Formally,
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144 for each pixel $(x,y)$ of the output image, we give the value of the pixel
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145 nearest to : $(ax+by+c,dx+ey+f)$, in the input image. This allows to
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146 produce scaling, translation, rotation and shearing variances.
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147
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148 The sampling of the parameters $(a,b,c,d,e,f)$ have been tuned by hand to
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149 forbid important rotations (not to confuse classes) but to give good
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150 variability of the transformation. For each image we sample uniformly the
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151 parameters in the following ranges: $a$ and $d$ in $[1-3 \times
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152 complexity,1+3 \times complexity]$, $b$ and $e$ in $[-3 \times complexity,3
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153 \times complexity]$ and $c$ and $f$ in $[-4 \times complexity, 4 \times
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154 complexity]$.
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155
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156
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157 \subsection{Local Elastic Deformations}
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158
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159 This filter induces a "wiggly" effect in the image. The description here
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160 will be brief, as the algorithm follows precisely what is described in
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161 \cite{SimardSP03}.
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162
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163 The general idea is to generate two "displacements" fields, for horizontal
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164 and vertical displacements of pixels. Each of these fields has the same
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165 size as the original image.
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166
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167 When generating the transformed image, we'll loop over the x and y
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168 positions in the fields and select, as a value, the value of the pixel in
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169 the original image at the (relative) position given by the displacement
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170 fields for this x and y. If the position we'd retrieve is outside the
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171 borders of the image, we use a 0 value instead.
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172
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173 To generate a pixel in either field, first a value between -1 and 1 is
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174 chosen from a uniform distribution. Then all the pixels, in both fields, is
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175 multiplied by a constant $\alpha$ which controls the intensity of the
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176 displacements (bigger $\alpha$ translates into larger wiggles).
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177
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178 As a final step, each field is convoluted with a Gaussian 2D kernel of
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179 standard deviation $\sigma$. Visually, this results in a "blur"
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180 filter. This has the effect of making values next to each other in the
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181 displacement fields similar. In effect, this makes the wiggles more
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182 coherent, less noisy.
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183
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184 As displacement fields were long to compute, 50 pairs of fields were
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185 generated per complexity in increments of 0.1 (50 pairs for 0.1, 50 pairs
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186 for 0.2, etc.), and afterwards, given a complexity, we selected randomly
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187 among the 50 corresponding pairs.
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188
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189 $\sigma$ and $\alpha$ were linked to complexity through the formulas
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190 $\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times
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191 \sqrt[3]{complexity}$.
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192
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193
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194 \subsection{Pinch}
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195
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196 This is another GIMP filter we used. The filter is in fact named "Whirl and
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197 pinch", but we don't use the "whirl" part (whirl is set to 0). As described
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198 in GIMP, a pinch is "similar to projecting the image onto an elastic
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199 surface and pressing or pulling on the center of the surface".
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200
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201 Mathematically, for a square input image, think of drawing a circle of
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202 radius $r$ around a center point $C$. Any point (pixel) $P$ belonging to
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203 that disk (region inside circle) will have its value recalculated by taking
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204 the value of another "source" pixel in the original image. The position of
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205 that source pixel is found on the line thats goes through $C$ and $P$, but
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206 at some other distance $d_2$. Define $d_1$ to be the distance between $P$
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207 and $C$. $d_2$ is given by $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times
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208 d_1$, where $pinch$ is a parameter to the filter.
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209
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210 If the region considered is not square then, before computing $d_2$, the
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211 smallest dimension (x or y) is stretched such that we may consider the
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212 region as if it was square. Then, after $d_2$ has been computed and
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213 corresponding components $d_2\_x$ and $d_2\_y$ have been found, the
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214 component corresponding to the stretched dimension is compressed back by an
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215 inverse ratio.
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216
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217 The actual value is given by bilinear interpolation considering the pixels
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218 around the (non-integer) source position thus found.
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219
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220 The value for $pinch$ in our case was given by sampling from an uniform
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221 distribution over the range $[-complexity, 0.7 \times complexity]$.
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222
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223 \subsection{Motion Blur}
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224
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225 This is a GIMP filter we applied, a "linear motion blur" in GIMP
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226 terminology. The description will be brief as it is a well-known filter.
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227
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228 This algorithm has two input parameters, $length$ and $angle$. The value of
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229 a pixel in the final image is the mean value of the $length$ first pixels
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230 found by moving in the $angle$ direction. An approximation of this idea is
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231 used, as we won't fall onto precise pixels by following that
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232 direction. This is done using the Bresenham line algorithm.
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233
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234 The angle, in our case, is chosen from a uniform distribution over
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235 $[0,360]$ degrees. The length, though, depends on the complexity; it's
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236 sampled from a Gaussian distribution of mean 0 and standard deviation
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237 $\sigma = 3 \times complexity$.
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238
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239 \subsection{Occlusion}
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240
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241 This filter selects random parts of other (hereafter "occlusive") letter
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242 images and places them over the original letter (hereafter "occluded")
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243 image. To be more precise, having selected a subregion of the occlusive
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244 image and a desination position in the occluded image, to determine the
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245 final value for a given overlapping pixel, it selects whichever pixel is
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246 the lightest. As a reminder, the background value is 0, black, so the value
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247 nearest to 1 is selected.
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248
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249 To select a subpart of the occlusive image, four numbers are generated. For
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250 compability with the code, we'll call them "haut", "bas", "gauche" and
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251 "droite" (respectively meaning top, bottom, left and right). Each of these
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252 numbers is selected according to a Gaussian distribution of mean $8 \times
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253 complexity$ and standard deviation $2$. This means the largest the
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254 complexity is, the biggest the occlusion will be. The absolute value is
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255 taken, as the numbers must be positive, and the maximum value is capped at
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256 15.
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257
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258 These four sizes collectively define a window centered on the middle pixel
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259 of the occlusive image. This is the part that will be extracted as the
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260 occlusion.
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261
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262 The next step is to select a destination position in the occluded
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263 image. Vertical and horizontal displacements $y\_arrivee$ and $x\_arrivee$
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264 are selected according to Gaussian distributions of mean 0 and of standard
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265 deviations of, respectively, 3 and 2. Then an horizontal placement mode,
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266 $place$, is selected to be of three values meaning
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267 left, middle or right.
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268
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269 If $place$ is "middle", the occlusion will be horizontally centered
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270 around the horizontal middle of the occluded image, then shifted according
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271 to $x\_arrivee$. If $place$ is "left", it will be placed on the left of
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272 the occluded image, then displaced right according to $x\_arrivee$. The
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273 contrary happens if $place$ is $right$.
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274
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275 In both the horizontal and vertical positionning, the maximum position in
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276 either direction is such that the selected occlusion won't go beyond the
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277 borders of the occluded image.
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278
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279 This filter has a probability of not being applied, at all, of 60\%.
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280
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281
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282 \subsection{Pixel Permutation}
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283
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284 This filter permuts neighbouring pixels. It selects first
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285 $\frac{complexity}{3}$ pixels randomly in the image. Each of them are then
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286 sequentially exchanged to one other pixel in its $V4$ neighbourhood. Number
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287 of exchanges to the left, right, top, bottom are equal or does not differ
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288 from more than 1 if the number of selected pixels is not a multiple of 4.
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289
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290 It has has a probability of not being applied, at all, of 80\%.
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291
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292
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293 \subsection{Gaussian Noise}
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294
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295 This filter simply adds, to each pixel of the image independently, a
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296 Gaussian noise of mean $0$ and standard deviation $\frac{complexity}{10}$.
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297
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298 It has has a probability of not being applied, at all, of 70\%.
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299
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300
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301 \subsection{Background Images}
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302
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303 Following~\cite{Larochelle-jmlr-2009}, this transformation adds a random
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304 background behind the letter. The background is chosen by first selecting,
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305 at random, an image from a set of images. Then we choose a 32x32 subregion
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306 of that image as the background image (by sampling x and y positions
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307 uniformly while making sure not to cross image borders).
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308
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309 To combine the original letter image and the background image, contrast
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310 adjustments are made. We first get the maximal values (i.e. maximal
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311 intensity) for both the original image and the background image, $maximage$
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312 and $maxbg$. We also have a parameter, $contrast$, given by sampling from a
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313 uniform distribution over $[complexity, 1]$.
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314
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315 Once we have all these numbers, we first adjust the values for the
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316 background image. Each pixel value is multiplied by $\frac{max(maximage -
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317 contrast, 0)}{maxbg}$. Therefore the higher the contrast, the darkest the
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318 background will be.
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319
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320 The final image is found by taking the brightest (i.e. value nearest to 1)
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321 pixel from either the background image or the corresponding pixel in the
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322 original image.
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323
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324 \subsection{Salt and Pepper Noise}
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325
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326 This filter adds noise to the image by randomly selecting a certain number
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327 of them and, for those selected pixels, assign a random value according to
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328 a uniform distribution over the $[0,1]$ ranges. This last distribution does
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329 not change according to complexity. Instead, the number of selected pixels
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330 does: the proportion of changed pixels corresponds to $complexity / 5$,
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331 which means, as a maximum, 20\% of the pixels will be randomized. On the
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332 lowest extreme, no pixel is changed.
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333
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334 This filter also has a probability of not being applied, at all, of 75\%.
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335
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336 \subsection{Spatially Gaussian Noise}
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337
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338 The aim of this transformation is to filter, with a gaussian kernel,
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339 different regions of the image. In order to save computing time we decided
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340 to convolve the whole image only once with a symmetric gaussian kernel of
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341 size and variance choosen uniformly in the ranges: $[12,12 + 20 \times
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342 complexity]$ and $[2,2 + 6 \times complexity]$. The result is normalized
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343 between $0$ and $1$. We also create a symmetric averaging window, of the
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344 kernel size, with maximum value at the center. For each image we sample
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345 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be
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346 averaging centers between the original image and the filtered one. We
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347 initialize to zero a mask matrix of the image size. For each selected pixel
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348 we add to the mask the averaging window centered to it. The final image is
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349 computed from the following element-wise operation: $\frac{image + filtered
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350 image \times mask}{mask+1}$.
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351
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352 This filter has a probability of not being applied, at all, of 75\%.
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353
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354 \subsection{Scratches}
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355
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356 The scratches module places line-like white patches on the image. The
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357 lines are in fact heavily transformed images of the digit "1" (one), chosen
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358 at random among five thousands such start images of this digit.
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359
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360 Once the image is selected, the transformation begins by finding the first
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361 $top$, $bottom$, $right$ and $left$ non-zero pixels in the image. It is
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362 then cropped to the region thus delimited, then this cropped version is
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363 expanded to $32\times32$ again. It is then rotated by a random angle having a
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364 Gaussian distribution of mean 90 and standard deviation $100 \times
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365 complexity$ (in degrees). The rotation is done with bicubic interpolation.
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366
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367 The rotated image is then resized to $50\times50$, with anti-aliasing. In
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368 that image, we crop the image again by selecting a region delimited
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369 horizontally to $left$ to $left+32$ and vertically by $top$ to $top+32$.
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370
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371 Once this is done, two passes of a greyscale morphological erosion filter
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372 are applied. Put briefly, this erosion filter reduces the width of the line
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373 by a certain $smoothing$ amount. For small complexities (< 0.5),
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374 $smoothing$ is 6, so the line is very small. For complexities ranging from
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375 0.25 to 0.5, $smoothing$ is 5. It is 4 for complexities 0.5 to 0.75, and 3
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376 for higher complexities.
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377
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378 To compensate for border effects, the image is then cropped to 28x28 by
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379 removing two pixels everywhere on the borders, then expanded to 32x32
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380 again. The pixel values are then linearly expanded such that the minimum
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381 value is 0 and the maximal one is 1. Then, 50\% of the time, the image is
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382 vertically flipped.
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383
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384 This filter is only applied only 15\% of the time. When it is applied, 50\%
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385 of the time, only one patch image is generated and applied. In 30\% of
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386 cases, two patches are generated, and otherwise three patches are
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387 generated. The patch is applied by taking the maximal value on any given
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388 patch or the original image, for each of the 32x32 pixel locations.
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389
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390 \subsection{Color and Contrast Changes}
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391
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392 This filter changes the constrast and may invert the image polarity (white
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393 on black to black on white). The contrast $C$ is defined here as the
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394 difference between the maximum and the minimum pixel value of the image. A
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395 contrast value is sampled uniformly between $1$ and $1-0.85 \times
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396 complexity$ (this insure a minimum constrast of $0.15$). We then simply
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397 normalize the image to the range $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The
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398 polarity is inverted with $0.5$ probability.
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399
379
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400
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401 \begin{figure}[h]
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402 \resizebox{.99\textwidth}{!}{\includegraphics{images/example_t.png}}\\
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403 \caption{Illustration of the pipeline of stochastic
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parents: 392
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404 transformations applied to the image of a lower-case t
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parents: 392
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405 (the upper left image). Each image in the pipeline (going from
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406 left to right, first top line, then bottom line) shows the result
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407 of applying one of the modules in the pipeline. The last image
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408 (bottom right) is used as training example.}
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409 \label{fig:pipeline}
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410 \end{figure}
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411
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412
479
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413 \begin{figure}[h]
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414 \resizebox{.99\textwidth}{!}{\includegraphics{images/transfo.png}}\\
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415 \caption{Illustration of each transformation applied to the same image
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416 of the upper-case h (upper-left image). first row (from left to rigth) : original image, slant,
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417 thickness, affine transformation, local elastic deformation; second row (from left to rigth) :
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418 pinch, motion blur, occlusion, pixel permutation, gaussian noise; third row (from left to rigth) :
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419 background image, salt and pepper noise, spatially gaussian noise, scratches,
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parents: 477
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420 color and contrast changes.}
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421 \label{fig:transfo}
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422 \end{figure}
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423
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424
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425 \section{Experimental Setup}
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426
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427 \subsection{Training Datasets}
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428
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429 \subsubsection{Data Sources}
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430
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431 \begin{itemize}
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432 \item {\bf NIST}
434
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433 The NIST Special Database 19 (NIST19) is a very widely used dataset for training and testing OCR systems.
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434 The dataset is composed with over 800 000 digits and characters (upper and lower cases), with hand checked classifications,
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435 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes
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436 corresponding to "0"-"9","A"-"Z" and "a"-"z". The dataset contains 8 series of different complexity.
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437 The fourth series, $hsf_4$, experimentally recognized to be the most difficult one for classification task is recommended
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438 by NIST as testing set and is used in our work for that purpose.
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439 The performances reported by previous work on that dataset mostly use only the digits.
432
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440 Here we use the whole classes both in the training and testing phase.
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441
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442
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443 \item {\bf Fonts}
477
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444 In order to have a good variety of sources we downloaded an important number of free fonts from: {\tt http://anonymous.url.net}
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445 %real adress {\tt http://cg.scs.carleton.ca/~luc/freefonts.html}
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446 in addition to Windows 7's, this adds up to a total of $9817$ different fonts that we can choose uniformly.
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447 The ttf file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image,
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448 directly as input to our models.
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449 %Guillaume are there other details I forgot on the font selection?
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450
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451 \item {\bf Captchas}
432
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452 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for
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453 generating characters of the same format as the NIST dataset. The core of this data source is composed with a random character
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454 generator and various kinds of tranformations similar to those described in the previous sections.
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455 In order to increase the variability of the data generated, different fonts are used for generating the characters.
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456 Transformations (slant, distorsions, rotation, translation) are applied to each randomly generated character with a complexity
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457 depending on the value of the complexity parameter provided by the user of the data source. Two levels of complexity are
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458 allowed and can be controlled via an easy to use facade class.
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459 \item {\bf OCR data}
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460 \end{itemize}
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461
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462 \subsubsection{Data Sets}
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463 \begin{itemize}
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464 \item {\bf P07}
420
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465 The dataset P07 is sampled with our transformation pipeline with a complexity parameter of $0.7$.
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466 For each new exemple to generate, we choose one source with the following probability: $0.1$ for the fonts,
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467 $0.25$ for the captchas, $0.25$ for OCR data and $0.4$ for NIST. We apply all the transformations in their order
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468 and for each of them we sample uniformly a complexity in the range $[0,0.7]$.
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469 \item {\bf NISTP} {\em ne pas utiliser PNIST mais NISTP, pour rester politically correct...}
463
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parents: 462
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470 NISTP is equivalent to P07 (complexity parameter of $0.7$ with the same sources proportion) except that we only apply transformations from slant to pinch. Therefore, the character is transformed
420
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471 but no additionnal noise is added to the image, this gives images closer to the NIST dataset.
379
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472 \end{itemize}
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473
452
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474 We noticed that the distribution of the training sets and the test sets differ.
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475 Since our validation sets are sampled from the training set, they have approximately the same distribution, but the test set has a completely different distribution as illustrated in figure \ref {setsdata}.
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476
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477 \begin{figure}
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478 \subfigure[NIST training]{\includegraphics[width=0.5\textwidth]{images/nisttrainstats}}
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479 \subfigure[NIST validation]{\includegraphics[width=0.5\textwidth]{images/nistvalidstats}}
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480 \subfigure[NIST test]{\includegraphics[width=0.5\textwidth]{images/nistteststats}}
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481 \subfigure[NISTP validation]{\includegraphics[width=0.5\textwidth]{images/nistpvalidstats}}
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482 \caption{Proportion of each class in some of the data sets}
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parents: 444
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483 \label{setsdata}
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484 \end{figure}
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485
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486 \subsection{Models and their Hyperparameters}
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487
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488 \subsubsection{Multi-Layer Perceptrons (MLP)}
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489
410
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490 An MLP is a family of functions that are described by stacking layers of of a function similar to
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491 $$g(x) = \tanh(b+Wx)$$
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492 The input, $x$, is a $d$-dimension vector.
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493 The output, $g(x)$, is a $m$-dimension vector.
411
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494 The parameter $W$ is a $m\times d$ matrix and is called the weight matrix.
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495 The parameter $b$ is a $m$-vector and is called the bias vector.
410
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496 The non-linearity (here $\tanh$) is applied element-wise to the output vector.
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parents: 407
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497 Usually the input is referred to a input layer and similarly for the output.
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498 You can of course chain several such functions to obtain a more complex one.
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parents: 407
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499 Here is a common example
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500 $$f(x) = c + V\tanh(b+Wx)$$
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parents: 407
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501 In this case the intermediate layer corresponding to $\tanh(b+Wx)$ is called a hidden layer.
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parents: 407
diff changeset
502 Here the output layer does not have the same non-linearity as the hidden layer.
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parents: 407
diff changeset
503 This is a common case where some specialized non-linearity is applied to the output layer only depending on the task at hand.
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parents: 407
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504
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parents: 407
diff changeset
505 If you put 3 or more hidden layers in such a network you obtain what is called a deep MLP.
411
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parents: 410
diff changeset
506 The parameters to adapt are the weight matrix and the bias vector for each layer.
410
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507
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508 \subsubsection{Stacked Denoising Auto-Encoders (SDAE)}
422
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parents: 417
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509 \label{SdA}
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510
410
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511 Auto-encoders are essentially a way to initialize the weights of the network to enable better generalization.
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512 This is essentially unsupervised training where the layer is made to reconstruct its input through and encoding and decoding phase.
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parents: 417
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513 Denoising auto-encoders are a variant where the input is corrupted with random noise but the target is the uncorrupted input.
410
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514 The principle behind these initialization methods is that the network will learn the inherent relation between portions of the data and be able to represent them thus helping with whatever task we want to perform.
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515
411
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516 An auto-encoder unit is formed of two MLP layers with the bottom one called the encoding layer and the top one the decoding layer.
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517 Usually the top and bottom weight matrices are the transpose of each other and are fixed this way.
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518 The network is trained as such and, when sufficiently trained, the MLP layer is initialized with the parameters of the encoding layer.
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519 The other parameters are discarded.
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520
410
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521 The stacked version is an adaptation to deep MLPs where you initialize each layer with a denoising auto-encoder starting from the bottom.
411
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diff changeset
522 During the initialization, which is usually called pre-training, the bottom layer is treated as if it were an isolated auto-encoder.
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523 The second and following layers receive the same treatment except that they take as input the encoded version of the data that has gone through the layers before it.
410
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524 For additional details see \cite{vincent:icml08}.
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525
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526 \section{Experimental Results}
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527
438
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528 \subsection{SDA vs MLP vs Humans}
392
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529
438
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530 We compare here the best MLP (according to validation set error) that we found against
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parents: 437
diff changeset
531 the best SDA (again according to validation set error), along with a precise estimate
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parents: 437
diff changeset
532 of human performance obtained via Amazon's Mechanical Turk (AMT)
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parents: 437
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533 service\footnote{http://mturk.com}. AMT users are paid small amounts
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parents: 437
diff changeset
534 of money to perform tasks for which human intelligence is required.
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parents: 437
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535 Mechanical Turk has been used extensively in natural language
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parents: 437
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536 processing \cite{SnowEtAl2008} and vision
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537 \cite{SorokinAndForsyth2008,whitehill09}. AMT users where presented
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diff changeset
538 with 10 character images and asked to type 10 corresponding ascii
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539 characters. Hence they were forced to make a hard choice among the
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540 62 character classes. Three users classified each image, allowing
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541 to estimate inter-human variability (shown as +/- in parenthesis below).
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542
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543 \begin{table}
458
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544 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits +
438
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545 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training
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546 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture
458
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547 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07)
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548 and using a validation set to select hyper-parameters and other training choices.
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549 \{SDA,MLP\}0 are trained on NIST,
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550 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07.
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551 The human error rate on digits is a lower bound because it does not count digits that were
461
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552 recognized as letters. For comparison, the results found in the literature
9609c5cf9b6b lit. results
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553 on NIST digits classification using the same test set are included.}
438
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554 \label{tab:sda-vs-mlp-vs-humans}
392
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555 \begin{center}
438
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556 \begin{tabular}{|l|r|r|r|r|} \hline
458
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557 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
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558 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $>1.1\%$ \\ \hline
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559 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline
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560 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline
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561 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline
460
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562 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline
458
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563 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline
460
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564 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline
461
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565 [5] & & & & 4.95\% $\pm$.18\% \\ \hline
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566 [2] & & & & 3.71\% $\pm$.16\% \\ \hline
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567 [3] & & & & 2.4\% $\pm$.13\% \\ \hline
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568 [4] & & & & 2.1\% $\pm$.12\% \\ \hline
392
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569 \end{tabular}
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570 \end{center}
438
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571 \end{table}
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572
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573 \subsection{Perturbed Training Data More Helpful for SDAE}
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574
460
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575 \begin{table}
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576 \caption{Relative change in error rates due to the use of perturbed training data,
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577 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models.
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578 A positive value indicates that training on the perturbed data helped for the
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parents: 458
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579 given test set (the first 3 columns on the 62-class tasks and the last one is
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parents: 458
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580 on the clean 10-class digits). Clearly, the deep learning models did benefit more
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581 from perturbed training data, even when testing on clean data, whereas the MLP
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582 trained on perturbed data performed worse on the clean digits and about the same
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583 on the clean characters. }
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584 \label{tab:sda-vs-mlp-vs-humans}
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585 \begin{center}
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586 \begin{tabular}{|l|r|r|r|r|} \hline
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587 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
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parents: 458
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588 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline
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parents: 458
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589 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline
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590 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline
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591 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline
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592 \end{tabular}
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593 \end{center}
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594 \end{table}
458
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595
460
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596
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597 \subsection{Multi-Task Learning Effects}
379
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598
460
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599 As previously seen, the SDA is better able to benefit from the
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parents: 458
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600 transformations applied to the data than the MLP. In this experiment we
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601 define three tasks: recognizing digits (knowing that the input is a digit),
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602 recognizing upper case characters (knowing that the input is one), and
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603 recognizing lower case characters (knowing that the input is one). We
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604 consider the digit classification task as the target task and we want to
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605 evaluate whether training with the other tasks can help or hurt, and
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606 whether the effect is different for MLPs versus SDAs. The goal is to find
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607 out if deep learning can benefit more (or less) from multiple related tasks
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608 (i.e. the multi-task setting) compared to a corresponding purely supervised
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609 shallow learner.
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610
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611 We use a single hidden layer MLP with 1000 hidden units, and a SDA
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612 with 3 hidden layers (1000 hidden units per layer), pre-trained and
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613 fine-tuned on NIST.
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614
460
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615 Our results show that the MLP benefits marginally from the multi-task setting
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616 in the case of digits (5\% relative improvement) but is actually hurt in the case
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617 of characters (respectively 3\% and 4\% worse for lower and upper class characters).
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618 On the other hand the SDA benefitted from the multi-task setting, with relative
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619 error rate improvements of 27\%, 15\% and 13\% respectively for digits,
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620 lower and upper case characters, as shown in Table~\ref{tab:multi-task}.
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621
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622 \begin{table}
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623 \caption{Test error rates and relative change in error rates due to the use of
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624 a multi-task setting, i.e., training on each task in isolation vs training
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625 for all three tasks together, for MLPs vs SDAs. The SDA benefits much
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626 more from the multi-task setting. All experiments on only on the
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627 unperturbed NIST data, using validation error for model selection.
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628 Relative improvement is 1 - single-task error / multi-task error.}
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629 \label{tab:multi-task}
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630 \begin{center}
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631 \begin{tabular}{|l|r|r|r|} \hline
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632 & single-task & multi-task & relative \\
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633 & setting & setting & improvement \\ \hline
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634 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline
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635 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline
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636 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline
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637 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline
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638 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline
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639 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline
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640 \end{tabular}
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641 \end{center}
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642 \end{table}
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643
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644 \section{Conclusions}
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645
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646 \bibliography{strings,ml,aigaion,specials}
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647 \bibliographystyle{mlapa}
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648
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649 \end{document}