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aistats submission looking good
author Yoshua Bengio <bengioy@iro.umontreal.ca>
date Sun, 31 Oct 2010 22:27:30 -0400
parents 1f5d2d01b84d
children eb6244c6d861
rev   line source
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1 %\documentclass[twoside,11pt]{article} % For LaTeX2e
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2 \documentclass{article} % For LaTeX2e
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3 \usepackage{aistats2e_2011}
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4 %\usepackage{times}
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5 \usepackage{wrapfig}
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6 \usepackage{amsthm}
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7 \usepackage{amsmath}
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8 \usepackage{bbm}
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9 \usepackage[utf8]{inputenc}
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10 \usepackage[psamsfonts]{amssymb}
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11 %\usepackage{algorithm,algorithmic} % not used after all
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12 \usepackage{graphicx,subfigure}
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13 \usepackage[numbers]{natbib}
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14
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15 \addtolength{\textwidth}{10mm}
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16 \addtolength{\evensidemargin}{-5mm}
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17 \addtolength{\oddsidemargin}{-5mm}
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18
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19 %\setlength\parindent{0mm}
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20
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21 \begin{document}
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22
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23 \twocolumn[
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24 \aistatstitle{Deeper Learners Benefit More from Multi-Task and Perturbed Examples}
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25 \runningtitle{Deep Learners for Out-of-Distribution Examples}
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26 \runningauthor{Bengio et. al.}
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27 \aistatsauthor{Anonymous Authors}]
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28 \iffalse
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29 Yoshua Bengio \and
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30 Frédéric Bastien \and
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31 Arnaud Bergeron \and
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32 Nicolas Boulanger-Lewandowski \and
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33 Thomas Breuel \and
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34 Youssouf Chherawala \and
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35 Moustapha Cisse \and
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36 Myriam Côté \and
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37 Dumitru Erhan \and
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38 Jeremy Eustache \and
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39 Xavier Glorot \and
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40 Xavier Muller \and
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41 Sylvain Pannetier Lebeuf \and
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42 Razvan Pascanu \and
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43 Salah Rifai \and
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44 Francois Savard \and
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45 Guillaume Sicard
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46 %}
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47 \fi
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48 %\aistatsaddress{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada}
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49 %\date{{\tt bengioy@iro.umontreal.ca}, Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada}
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50 %\jmlrheading{}{2010}{}{10/2010}{XX/2011}{Yoshua Bengio et al}
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51 %\editor{}
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52
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53 %\makeanontitle
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54 %\maketitle
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55
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56 %{\bf Running title: Deep Self-Taught Learning}
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57
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58 %\vspace*{-2mm}
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59 \begin{abstract}
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60 Recent theoretical and empirical work in statistical machine learning has demonstrated the potential of learning algorithms for deep architectures, i.e., function classes obtained by composing multiple levels of representation. The hypothesis evaluated here is that intermediate levels of representation, because
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61 they can be shared across tasks and examples from different but related
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62 distributions, can yield even more benefits where there are more such levels of representation. The experiments are performed on a large-scale handwritten character recognition setting with 62 classes (upper case, lower case, digits). We show that a deep learner could not only {\em beat previously published results but also reach human-level performance}.
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63 \end{abstract}
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64 %\vspace*{-3mm}
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65
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66 %\begin{keywords}
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67 %Deep learning, self-taught learning, out-of-distribution examples, handwritten character recognition, multi-task learning
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68 %\end{keywords}
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69 %\keywords{self-taught learning \and multi-task learning \and out-of-distribution examples \and deep learning \and handwriting recognition}
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70
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71
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72
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73 \section{Introduction}
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74 %\vspace*{-1mm}
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75
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76 {\bf Deep Learning} has emerged as a promising new area of research in
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77 statistical machine learning~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,VincentPLarochelleH2008,ranzato-08,TaylorHintonICML2009,Larochelle-jmlr-2009,Salakhutdinov+Hinton-2009,HonglakL2009,HonglakLNIPS2009,Jarrett-ICCV2009,Taylor-cvpr-2010}. See \citet{Bengio-2009} for a review.
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78 Learning algorithms for deep architectures are centered on the learning
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79 of useful representations of data, which are better suited to the task at hand,
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80 and are organized in a hierarchy with multiple levels.
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81 This is in part inspired by observations of the mammalian visual cortex,
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82 which consists of a chain of processing elements, each of which is associated with a
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83 different representation of the raw visual input. In fact,
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84 it was found recently that the features learnt in deep architectures resemble
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85 those observed in the first two of these stages (in areas V1 and V2
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86 of visual cortex) \citep{HonglakL2008}, and that they become more and
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87 more invariant to factors of variation (such as camera movement) in
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88 higher layers~\citep{Goodfellow2009}.
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89 Learning a hierarchy of features increases the
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90 ease and practicality of developing representations that are at once
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91 tailored to specific tasks, yet are able to borrow statistical strength
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92 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
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93 feature representation can lead to higher-level (more abstract, more
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94 general) features that are more robust to unanticipated sources of
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95 variance extant in real data.
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96
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97 Whereas a deep architecture can in principle be more powerful than a
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98 shallow one in terms of representation, depth appears to render the
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99 training problem more difficult in terms of optimization and local minima.
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100 It is also only recently that successful algorithms were proposed to
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101 overcome some of these difficulties. All are based on unsupervised
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102 learning, often in an greedy layer-wise ``unsupervised pre-training''
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103 stage~\citep{Bengio-2009}.
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104 The principle is that each layer starting from
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105 the bottom is trained to represent its input (the output of the previous
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106 layer). After this
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107 unsupervised initialization, the stack of layers can be
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108 converted into a deep supervised feedforward neural network and fine-tuned by
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109 stochastic gradient descent.
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110 One of these layer initialization techniques,
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111 applied here, is the Denoising
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112 Auto-encoder~(DAE)~\citep{VincentPLarochelleH2008-very-small} (see
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113 Figure~\ref{fig:da}), which performed similarly or
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114 better~\citep{VincentPLarochelleH2008-very-small} than previously
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115 proposed Restricted Boltzmann Machines (RBM)~\citep{Hinton06}
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116 in terms of unsupervised extraction
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117 of a hierarchy of features useful for classification. Each layer is trained
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118 to denoise its input, creating a layer of features that can be used as
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119 input for the next layer. Note that training a Denoising Auto-Encoder
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120 can actually been seen as training a particular RBM by an inductive
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121 principle different from maximum likelihood~\cite{Vincent-SM-2010}, namely by
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122 Score Matching~\citep{Hyvarinen-2005,HyvarinenA2008}.
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123
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124 Previous comparative experimental results with stacking of RBMs and DAEs
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125 to build deep supervised predictors had shown that they could outperform
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126 shallow architectures in a variety of settings (see~\citet{Bengio-2009}
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127 for a review), especially
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128 when the data involves complex interactions between many factors of
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129 variation~\citep{LarochelleH2007}. Other experiments have suggested
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130 that the unsupervised layer-wise pre-training acted as a useful
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131 prior~\citep{Erhan+al-2010} that allows one to initialize a deep
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132 neural network in a relatively much smaller region of parameter space,
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133 corresponding to better generalization.
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134
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135 To further the understanding of the reasons for the good performance
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136 observed with deep learners, we focus here on the following {\em hypothesis}:
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137 intermediate levels of representation, especially when there are
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138 more such levels, can be exploited to {\bf share
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139 statistical strength across different but related types of examples},
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140 such as examples coming from other tasks than the task of interest
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141 (the multi-task setting), or examples coming from an overlapping
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142 but different distribution (images with different kinds of perturbations
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143 and noises, here). This is consistent with the hypotheses discussed
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144 at length in~\citet{Bengio-2009} regarding the potential advantage
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145 of deep learning and the idea that more levels of representation can
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146 give rise to more abstract, more general features of the raw input.
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147
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148 This hypothesis is related to a learning setting called
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149 {\bf self-taught learning}~\citep{RainaR2007}, which combines principles
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150 of semi-supervised and multi-task learning: the learner can exploit examples
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151 that are unlabeled and possibly come from a distribution different from the target
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152 distribution, e.g., from other classes than those of interest.
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153 It has already been shown that deep learners can clearly take advantage of
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154 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small},
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155 but more needed to be done to explore the impact
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156 of {\em out-of-distribution} examples and of the {\em multi-task} setting
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157 (one exception is~\citep{CollobertR2008}, which shares and uses unsupervised
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158 pre-training only with the first layer). In particular the {\em relative
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159 advantage of deep learning} for these settings has not been evaluated.
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160
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161
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162 %
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163 The {\bf main claim} of this paper is that deep learners (with several levels of representation) can
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164 {\bf benefit more from self-taught learning than shallow learners} (with a single
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165 level), both in the context of the multi-task setting and from {\em
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166 out-of-distribution examples} in general. Because we are able to improve on state-of-the-art
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167 performance and reach human-level performance
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168 on a large-scale task, we consider that this paper is also a contribution
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169 to advance the application of machine learning to handwritten character recognition.
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170 More precisely, we ask and answer the following questions:
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171
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172 %\begin{enumerate}
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173 $\bullet$ %\item
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174 Do the good results previously obtained with deep architectures on the
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175 MNIST digit images generalize to the setting of a similar but much larger and richer
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176 dataset, the NIST special database 19, with 62 classes and around 800k examples?
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177
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178 $\bullet$ %\item
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179 To what extent does the perturbation of input images (e.g. adding
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180 noise, affine transformations, background images) make the resulting
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181 classifiers better not only on similarly perturbed images but also on
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182 the {\em original clean examples}? We study this question in the
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183 context of the 62-class and 10-class tasks of the NIST special database 19.
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184
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185 $\bullet$ %\item
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186 Do deep architectures {\em benefit {\bf more} from such out-of-distribution}
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187 examples, in particular do they benefit more from
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188 examples that are perturbed versions of the examples from the task of interest?
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189
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190 $\bullet$ %\item
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191 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more}
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192 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than
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193 a corresponding shallow and purely supervised architecture?
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194 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case)
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195 to answer this question.
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196 %\end{enumerate}
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197
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198 Our experimental results provide positive evidence towards all of these questions,
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199 as well as {\em classifiers that reach human-level performance on 62-class isolated character
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200 recognition and beat previously published results on the NIST dataset (special database 19)}.
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201 To achieve these results, we introduce in the next section a sophisticated system
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202 for stochastically transforming character images and then explain the methodology,
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203 which is based on training with or without these transformed images and testing on
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204 clean ones. We measure the relative advantage of out-of-distribution examples
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205 (perturbed or out-of-class)
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206 for a deep learner vs a supervised shallow one.
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207 Code for generating these transformations as well as for the deep learning
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208 algorithms are made available at {\tt http://anonymous.url.net}.%{\tt http://hg.assembla.com/ift6266}.
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209 We also estimate the relative advantage for deep learners of training with
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210 other classes than those of interest, by comparing learners trained with
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211 62 classes with learners trained with only a subset (on which they
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212 are then tested).
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213 The conclusion discusses
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214 the more general question of why deep learners may benefit so much from
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215 the self-taught learning framework. Since out-of-distribution data
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216 (perturbed or from other related classes) is very common, this conclusion
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217 is of practical importance.
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218
602
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219 %\vspace*{-3mm}
600
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220 %\newpage
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221 \section{Perturbed and Transformed Character Images}
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222 \label{s:perturbations}
602
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223 %\vspace*{-2mm}
600
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224
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225 Figure~\ref{fig:transform} shows the different transformations we used to stochastically
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226 transform $32 \times 32$ source images (such as the one in Fig.\ref{fig:torig})
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227 in order to obtain data from a larger distribution which
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228 covers a domain substantially larger than the clean characters distribution from
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229 which we start.
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230 Although character transformations have been used before to
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231 improve character recognizers, this effort is on a large scale both
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232 in number of classes and in the complexity of the transformations, hence
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233 in the complexity of the learning task.
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234 The code for these transformations (mostly python) is available at
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235 {\tt http://anonymous.url.net}. All the modules in the pipeline share
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236 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
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237 amount of deformation or noise introduced.
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238 There are two main parts in the pipeline. The first one,
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239 from slant to pinch below, performs transformations. The second
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240 part, from blur to contrast, adds different kinds of noise.
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241 More details can be found in~\citep{ift6266-tr-anonymous}.
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242
602
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243 \begin{figure*}[ht]
600
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244 \centering
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245 \subfigure[Original]{\includegraphics[scale=0.6]{images/Original.png}\label{fig:torig}}
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246 \subfigure[Thickness]{\includegraphics[scale=0.6]{images/Thick_only.png}}
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247 \subfigure[Slant]{\includegraphics[scale=0.6]{images/Slant_only.png}}
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248 \subfigure[Affine Transformation]{\includegraphics[scale=0.6]{images/Affine_only.png}}
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249 \subfigure[Local Elastic Deformation]{\includegraphics[scale=0.6]{images/Localelasticdistorsions_only.png}}
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250 \subfigure[Pinch]{\includegraphics[scale=0.6]{images/Pinch_only.png}}
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251 %Noise
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252 \subfigure[Motion Blur]{\includegraphics[scale=0.6]{images/Motionblur_only.png}}
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253 \subfigure[Occlusion]{\includegraphics[scale=0.6]{images/occlusion_only.png}}
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254 \subfigure[Gaussian Smoothing]{\includegraphics[scale=0.6]{images/Bruitgauss_only.png}}
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255 \subfigure[Pixels Permutation]{\includegraphics[scale=0.6]{images/Permutpixel_only.png}}
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256 \subfigure[Gaussian Noise]{\includegraphics[scale=0.6]{images/Distorsiongauss_only.png}}
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257 \subfigure[Background Image Addition]{\includegraphics[scale=0.6]{images/background_other_only.png}}
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258 \subfigure[Salt \& Pepper]{\includegraphics[scale=0.6]{images/Poivresel_only.png}}
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259 \subfigure[Scratches]{\includegraphics[scale=0.6]{images/Rature_only.png}}
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260 \subfigure[Grey Level \& Contrast]{\includegraphics[scale=0.6]{images/Contrast_only.png}}
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261 \caption{Top left (a): example original image. Others (b-o): examples of the effect
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262 of each transformation module taken separately. Actual perturbed examples are obtained by
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263 a pipeline of these, with random choices about which module to apply and how much perturbation
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264 to apply.}
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265 \label{fig:transform}
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266 %\vspace*{-2mm}
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267 \end{figure*}
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268
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269 %\vspace*{-3mm}
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270 \section{Experimental Setup}
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271 %\vspace*{-1mm}
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272
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273 Much previous work on deep learning had been performed on
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274 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009},
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275 with 60~000 examples, and variants involving 10~000
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276 examples~\citep{Larochelle-jmlr-2009,VincentPLarochelleH2008}.
600
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277 The focus here is on much larger training sets, from 10 times to
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278 to 1000 times larger, and 62 classes.
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279
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280 The first step in constructing the larger datasets (called NISTP and P07) is to sample from
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281 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas},
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282 and {\bf OCR data} (scanned machine printed characters). Once a character
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283 is sampled from one of these sources (chosen randomly), the second step is to
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284 apply a pipeline of transformations and/or noise processes outlined in section \ref{s:perturbations}.
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285
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286 To provide a baseline of error rate comparison we also estimate human performance
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287 on both the 62-class task and the 10-class digits task.
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288 We compare the best Multi-Layer Perceptrons (MLP) against
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289 the best Stacked Denoising Auto-encoders (SDA), when
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290 both models' hyper-parameters are selected to minimize the validation set error.
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291 We also provide a comparison against a precise estimate
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292 of human performance obtained via Amazon's Mechanical Turk (AMT)
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293 service ({\tt http://mturk.com}).
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294 AMT users are paid small amounts
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295 of money to perform tasks for which human intelligence is required.
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296 Mechanical Turk has been used extensively in natural language processing and vision.
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297 %processing \citep{SnowEtAl2008} and vision
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298 %\citep{SorokinAndForsyth2008,whitehill09}.
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299 AMT users were presented
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300 with 10 character images (from a test set) and asked to choose 10 corresponding ASCII
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301 characters. They were forced to choose a single character class (either among the
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302 62 or 10 character classes) for each image.
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303 80 subjects classified 2500 images per (dataset,task) pair.
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304 Different humans labelers sometimes provided a different label for the same
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305 example, and we were able to estimate the error variance due to this effect
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306 because each image was classified by 3 different persons.
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307 The average error of humans on the 62-class task NIST test set
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308 is 18.2\%, with a standard error of 0.1\%.
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309
602
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310 %\vspace*{-3mm}
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311 \subsection{Data Sources}
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312 %\vspace*{-2mm}
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313
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314 %\begin{itemize}
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315 %\item
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316 {\bf NIST.}
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317 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995},
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318 widely used for training and testing character
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319 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}.
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320 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications,
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321 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes
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322 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity.
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323 The fourth partition (called $hsf_4$, 82587 examples),
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324 experimentally recognized to be the most difficult one, is the one recommended
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325 by NIST as a testing set and is used in our work as well as some previous work~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}
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326 for that purpose. We randomly split the remainder (731668 examples) into a training set and a validation set for
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327 model selection.
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328 The performances reported by previous work on that dataset mostly use only the digits.
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329 Here we use all the classes both in the training and testing phase. This is especially
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330 useful to estimate the effect of a multi-task setting.
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331 The distribution of the classes in the NIST training and test sets differs
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332 substantially, with relatively many more digits in the test set, and a more uniform distribution
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333 of letters in the test set (whereas in the training set they are distributed
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334 more like in natural text).
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335 %\vspace*{-1mm}
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336
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337 %\item
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338 {\bf Fonts.}
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339 In order to have a good variety of sources we downloaded an important number of free fonts from:
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340 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}.
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341 % TODO: pointless to anonymize, it's not pointing to our work
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342 Including the operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from.
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343 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image,
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344 directly as input to our models.
602
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345 %\vspace*{-1mm}
600
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346
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347 %\item
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348 {\bf Captchas.}
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349 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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350 generating characters of the same format as the NIST dataset. This software is based on
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351 a random character class generator and various kinds of transformations similar to those described in the previous sections.
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352 In order to increase the variability of the data generated, many different fonts are used for generating the characters.
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353 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity
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354 depending on the value of the complexity parameter provided by the user of the data source.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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355 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class?
602
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356 %\vspace*{-1mm}
600
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357
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358 %\item
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359 {\bf OCR data.}
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360 A large set (2 million) of scanned, OCRed and manually verified machine-printed
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361 characters where included as an
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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362 additional source. This set is part of a larger corpus being collected by the Image Understanding
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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363 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern
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364 ({\tt http://www.iupr.com}), and which will be publicly released.
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365 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this
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366 %\end{itemize}
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367
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368 %\vspace*{-3mm}
600
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369 \subsection{Data Sets}
602
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370 %\vspace*{-2mm}
600
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371
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372 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label
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373 from one of the 62 character classes.
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374 %\begin{itemize}
602
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375 %\vspace*{-1mm}
600
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
376
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
377 %\item
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
378 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
379 \{651668 / 80000 / 82587\} \{training / validation / test\} examples.
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
380 %\vspace*{-1mm}
600
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
381
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
382 %\item
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
383 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
384 and sending them through the transformation pipeline described in section \ref{s:perturbations}.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
385 For each new example to generate, a data source is selected with probability $10\%$ from the fonts,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
386 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
387 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
388 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples.
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
389 %\vspace*{-1mm}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
390
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
391 %\item
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
392 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
393 except that we only apply
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
394 transformations from slant to pinch. Therefore, the character is
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
395 transformed but no additional noise is added to the image, giving images
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
396 closer to the NIST dataset.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
397 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
398 %\end{itemize}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
399
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
400 \begin{figure*}[ht]
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
401 %\vspace*{-2mm}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
402 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
403 %\vspace*{-2mm}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
404 \caption{Illustration of the computations and training criterion for the denoising
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
405 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
406 the layer (i.e. raw input or output of previous layer)
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
407 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$.
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
408 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
409 is compared to the uncorrupted input $x$ through the loss function
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
410 $L_H(x,z)$, whose expected value is approximately minimized during training
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
411 by tuning $\theta$ and $\theta'$.}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
412 \label{fig:da}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
413 %\vspace*{-2mm}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
414 \end{figure*}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
415
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
416 %\vspace*{-3mm}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
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417 \subsection{Models and their Hyperparameters}
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
418 %\vspace*{-2mm}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
419
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
420 The experiments are performed using MLPs (with a single
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
421 hidden layer) and SDAs.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
422 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
423
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
424 {\bf Multi-Layer Perceptrons (MLP).}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
425 Whereas previous work had compared deep architectures to both shallow MLPs and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
426 SVMs, we only compared to MLPs here because of the very large datasets used
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
427 (making the use of SVMs computationally challenging because of their quadratic
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
428 scaling behavior). Preliminary experiments on training SVMs (libSVM) with subsets of the training
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
429 set allowing the program to fit in memory yielded substantially worse results
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
430 than those obtained with MLPs. For training on nearly a billion examples
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
431 (with the perturbed data), the MLPs and SDA are much more convenient than
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
432 classifiers based on kernel methods.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
433 The MLP has a single hidden layer with $\tanh$ activation functions, and softmax (normalized
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
434 exponentials) on the output layer for estimating $P(class | image)$.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
435 The number of hidden units is taken in $\{300,500,800,1000,1500\}$.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
436 Training examples are presented in minibatches of size 20. A constant learning
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
437 rate was chosen among $\{0.001, 0.01, 0.025, 0.075, 0.1, 0.5\}$.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
438 %through preliminary experiments (measuring performance on a validation set),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
439 %and $0.1$ (which was found to work best) was then selected for optimizing on
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
440 %the whole training sets.
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
441 %\vspace*{-1mm}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
442
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
443
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
444 {\bf Stacked Denoising Auto-Encoders (SDA).}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
445 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
446 can be used to initialize the weights of each layer of a deep MLP (with many hidden
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
447 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006},
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
448 apparently setting parameters in the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
449 basin of attraction of supervised gradient descent yielding better
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
450 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
451 pre-training phase} uses all of the training images but not the training labels.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
452 Each layer is trained in turn to produce a new representation of its input
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
453 (starting from the raw pixels).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
454 It is hypothesized that the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
455 advantage brought by this procedure stems from a better prior,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
456 on the one hand taking advantage of the link between the input
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
457 distribution $P(x)$ and the conditional distribution of interest
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
458 $P(y|x)$ (like in semi-supervised learning), and on the other hand
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
459 taking advantage of the expressive power and bias implicit in the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
460 deep architecture (whereby complex concepts are expressed as
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
461 compositions of simpler ones through a deep hierarchy).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
462
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
463 Here we chose to use the Denoising
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
464 Auto-encoder~\citep{VincentPLarochelleH2008} as the building block for
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
465 these deep hierarchies of features, as it is simple to train and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
466 explain (see Figure~\ref{fig:da}, as well as
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
467 tutorial and code there: {\tt http://deeplearning.net/tutorial}),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
468 provides efficient inference, and yielded results
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
469 comparable or better than RBMs in series of experiments
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
470 \citep{VincentPLarochelleH2008}. During training, a Denoising
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
471 Auto-encoder is presented with a stochastically corrupted version
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
472 of the input and trained to reconstruct the uncorrupted input,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
473 forcing the hidden units to represent the leading regularities in
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
474 the data. Here we use the random binary masking corruption
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
475 (which sets to 0 a random subset of the inputs).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
476 Once it is trained, in a purely unsupervised way,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
477 its hidden units' activations can
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
478 be used as inputs for training a second one, etc.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
479 After this unsupervised pre-training stage, the parameters
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
480 are used to initialize a deep MLP, which is fine-tuned by
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
481 the same standard procedure used to train them (see previous section).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
482 The SDA hyper-parameters are the same as for the MLP, with the addition of the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
483 amount of corruption noise (we used the masking noise process, whereby a
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
484 fixed proportion of the input values, randomly selected, are zeroed), and a
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
485 separate learning rate for the unsupervised pre-training stage (selected
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
486 from the same above set). The fraction of inputs corrupted was selected
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
487 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
488 of hidden layers but it was fixed to 3 based on previous work with
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
489 SDAs on MNIST~\citep{VincentPLarochelleH2008}. The size of the hidden
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
490 layers was kept constant across hidden layers, and the best results
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
491 were obtained with the largest values that we could experiment
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
492 with given our patience, with 1000 hidden units.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
493
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
494 %\vspace*{-1mm}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
495
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
496 \begin{figure*}[ht]
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
497 %\vspace*{-2mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
498 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
499 %\vspace*{-3mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
500 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
501 on NIST, 1 on NISTP, and 2 on P07. Left: overall results
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
502 of all models, on NIST and NISTP test sets.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
503 Right: error rates on NIST test digits only, along with the previous results from
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
504 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
505 respectively based on ART, nearest neighbors, MLPs, and SVMs.}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
506 \label{fig:error-rates-charts}
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
507 %\vspace*{-2mm}
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
508 \end{figure*}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
509
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
510
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
511 \begin{figure*}[ht]
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
512 \vspace*{-3mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
513 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
514 \vspace*{-3mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
515 \caption{Relative improvement in error rate due to self-taught learning.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
516 Left: Improvement (or loss, when negative)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
517 induced by out-of-distribution examples (perturbed data).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
518 Right: Improvement (or loss, when negative) induced by multi-task
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
519 learning (training on all classes and testing only on either digits,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
520 upper case, or lower-case). The deep learner (SDA) benefits more from
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
521 both self-taught learning scenarios, compared to the shallow MLP.}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
522 \label{fig:improvements-charts}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
523 \vspace*{-2mm}
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
524 \end{figure*}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
525
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
526 \vspace*{-2mm}
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
527 \section{Experimental Results}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
528 \vspace*{-2mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
529
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
530 %%\vspace*{-1mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
531 %\subsection{SDA vs MLP vs Humans}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
532 %%\vspace*{-1mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
533 The models are either trained on NIST (MLP0 and SDA0),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
534 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
535 on either NIST, NISTP or P07, either on the 62-class task
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
536 or on the 10-digits task. Training (including about half
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
537 for unsupervised pre-training, for DAs) on the larger
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
538 datasets takes around one day on a GPU-285.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
539 Figure~\ref{fig:error-rates-charts} summarizes the results obtained,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
540 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
541 SDA2), along with the previous results on the digits NIST special database
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
542 19 test set from the literature, respectively based on ARTMAP neural
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
543 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
544 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
545 ~\citep{Milgram+al-2005}.% More detailed and complete numerical results
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
546 %(figures and tables, including standard errors on the error rates) can be
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
547 %found in Appendix.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
548 The deep learner not only outperformed the shallow ones and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
549 previously published performance (in a statistically and qualitatively
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
550 significant way) but when trained with perturbed data
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
551 reaches human performance on both the 62-class task
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
552 and the 10-class (digits) task.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
553 17\% error (SDA1) or 18\% error (humans) may seem large but a large
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
554 majority of the errors from humans and from SDA1 are from out-of-context
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
555 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
556 ``c'' and a ``C'' are often indistinguishible).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
557
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
558 In addition, as shown in the left of
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
559 Figure~\ref{fig:improvements-charts}, the relative improvement in error
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
560 rate brought by self-taught learning is greater for the SDA, and these
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
561 differences with the MLP are statistically and qualitatively
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
562 significant.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
563 The left side of the figure shows the improvement to the clean
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
564 NIST test set error brought by the use of out-of-distribution examples
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
565 (i.e. the perturbed examples examples from NISTP or P07).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
566 Relative percent change is measured by taking
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
567 $100 \% \times$ (original model's error / perturbed-data model's error - 1).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
568 The right side of
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
569 Figure~\ref{fig:improvements-charts} shows the relative improvement
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
570 brought by the use of a multi-task setting, in which the same model is
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
571 trained for more classes than the target classes of interest (i.e. training
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
572 with all 62 classes when the target classes are respectively the digits,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
573 lower-case, or upper-case characters). Again, whereas the gain from the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
574 multi-task setting is marginal or negative for the MLP, it is substantial
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
575 for the SDA. Note that to simplify these multi-task experiments, only the original
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
576 NIST dataset is used. For example, the MLP-digits bar shows the relative
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
577 percent improvement in MLP error rate on the NIST digits test set
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
578 is $100\% \times$ (single-task
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
579 model's error / multi-task model's error - 1). The single-task model is
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
580 trained with only 10 outputs (one per digit), seeing only digit examples,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
581 whereas the multi-task model is trained with 62 outputs, with all 62
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
582 character classes as examples. Hence the hidden units are shared across
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
583 all tasks. For the multi-task model, the digit error rate is measured by
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
584 comparing the correct digit class with the output class associated with the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
585 maximum conditional probability among only the digit classes outputs. The
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
586 setting is similar for the other two target classes (lower case characters
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
587 and upper case characters).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
588 %%\vspace*{-1mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
589 %\subsection{Perturbed Training Data More Helpful for SDA}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
590 %%\vspace*{-1mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
591
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
592 %%\vspace*{-1mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
593 %\subsection{Multi-Task Learning Effects}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
594 %%\vspace*{-1mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
595
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
596 \iffalse
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
597 As previously seen, the SDA is better able to benefit from the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
598 transformations applied to the data than the MLP. In this experiment we
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
599 define three tasks: recognizing digits (knowing that the input is a digit),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
600 recognizing upper case characters (knowing that the input is one), and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
601 recognizing lower case characters (knowing that the input is one). We
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
602 consider the digit classification task as the target task and we want to
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
603 evaluate whether training with the other tasks can help or hurt, and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
604 whether the effect is different for MLPs versus SDAs. The goal is to find
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
605 out if deep learning can benefit more (or less) from multiple related tasks
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
606 (i.e. the multi-task setting) compared to a corresponding purely supervised
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
607 shallow learner.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
608
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
609 We use a single hidden layer MLP with 1000 hidden units, and a SDA
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
610 with 3 hidden layers (1000 hidden units per layer), pre-trained and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
611 fine-tuned on NIST.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
612
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
613 Our results show that the MLP benefits marginally from the multi-task setting
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
614 in the case of digits (5\% relative improvement) but is actually hurt in the case
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
615 of characters (respectively 3\% and 4\% worse for lower and upper class characters).
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
616 On the other hand the SDA benefited from the multi-task setting, with relative
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
617 error rate improvements of 27\%, 15\% and 13\% respectively for digits,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
618 lower and upper case characters, as shown in Table~\ref{tab:multi-task}.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
619 \fi
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
620
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
621
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
622 \vspace*{-2mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
623 \section{Conclusions and Discussion}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
624 \vspace*{-2mm}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
625
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
626 We have found that the self-taught learning framework is more beneficial
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
627 to a deep learner than to a traditional shallow and purely
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
628 supervised learner. More precisely,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
629 the answers are positive for all the questions asked in the introduction.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
630 %\begin{itemize}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
631
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
632 $\bullet$ %\item
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
633 {\bf Do the good results previously obtained with deep architectures on the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
634 MNIST digits generalize to a much larger and richer (but similar)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
635 dataset, the NIST special database 19, with 62 classes and around 800k examples}?
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
636 Yes, the SDA {\em systematically outperformed the MLP and all the previously
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
637 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
638 performance} at around 17\% error on the 62-class task and 1.4\% on the digits,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
639 and beating previously published results on the same data.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
640
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
641 $\bullet$ %\item
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
642 {\bf To what extent do self-taught learning scenarios help deep learners,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
643 and do they help them more than shallow supervised ones}?
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
644 We found that distorted training examples not only made the resulting
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
645 classifier better on similarly perturbed images but also on
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
646 the {\em original clean examples}, and more importantly and more novel,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
647 that deep architectures benefit more from such {\em out-of-distribution}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
648 examples. MLPs were helped by perturbed training examples when tested on perturbed input
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
649 images (65\% relative improvement on NISTP)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
650 but only marginally helped (5\% relative improvement on all classes)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
651 or even hurt (10\% relative loss on digits)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
652 with respect to clean examples . On the other hand, the deep SDAs
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
653 were significantly boosted by these out-of-distribution examples.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
654 Similarly, whereas the improvement due to the multi-task setting was marginal or
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
655 negative for the MLP (from +5.6\% to -3.6\% relative change),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
656 it was quite significant for the SDA (from +13\% to +27\% relative change),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
657 which may be explained by the arguments below.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
658 %\end{itemize}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
659
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
660 In the original self-taught learning framework~\citep{RainaR2007}, the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
661 out-of-sample examples were used as a source of unsupervised data, and
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
662 experiments showed its positive effects in a \emph{limited labeled data}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
663 scenario. However, many of the results by \citet{RainaR2007} (who used a
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
664 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
665 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
666 We note instead that, for deep
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
667 architectures, our experiments show that such a positive effect is accomplished
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
668 even in a scenario with a \emph{large number of labeled examples},
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
669 i.e., here, the relative gain of self-taught learning is probably preserved
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
670 in the asymptotic regime.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
671
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
672 {\bf Why would deep learners benefit more from the self-taught learning framework}?
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
673 The key idea is that the lower layers of the predictor compute a hierarchy
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
674 of features that can be shared across tasks or across variants of the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
675 input distribution. A theoretical analysis of generalization improvements
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
676 due to sharing of intermediate features across tasks already points
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
677 towards that explanation~\cite{baxter95a}.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
678 Intermediate features that can be used in different
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
679 contexts can be estimated in a way that allows to share statistical
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
680 strength. Features extracted through many levels are more likely to
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
681 be more abstract and more invariant to some of the factors of variation
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
682 in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest),
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
683 increasing the likelihood that they would be useful for a larger array
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
684 of tasks and input conditions.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
685 Therefore, we hypothesize that both depth and unsupervised
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
686 pre-training play a part in explaining the advantages observed here, and future
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
687 experiments could attempt at teasing apart these factors.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
688 And why would deep learners benefit from the self-taught learning
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
689 scenarios even when the number of labeled examples is very large?
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
690 We hypothesize that this is related to the hypotheses studied
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
691 in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
692 it was found that online learning on a huge dataset did not make the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
693 advantage of the deep learning bias vanish, and a similar phenomenon
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
694 may be happening here. We hypothesize that unsupervised pre-training
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
695 of a deep hierarchy with self-taught learning initializes the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
696 model in the basin of attraction of supervised gradient descent
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
697 that corresponds to better generalization. Furthermore, such good
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
698 basins of attraction are not discovered by pure supervised learning
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
699 (with or without self-taught settings) from random initialization, and more labeled examples
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
700 does not allow the shallow or purely supervised models to discover
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
701 the kind of better basins associated
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
702 with deep learning and self-taught learning.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
703
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
704 A Flash demo of the recognizer (where both the MLP and the SDA can be compared)
602
203c6071e104 aistats submission looking good
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 600
diff changeset
705 can be executed on-line at the anonymous site {\tt http://deep.host22.com}.
600
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
706
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
707 \iffalse
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
708 \section*{Appendix I: Detailed Numerical Results}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
709
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
710 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
711 They also contain additional data such as test errors on P07 and standard errors.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
712
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
713 \begin{table}[ht]
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
714 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits +
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
715 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
716 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
717 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07)
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
718 and using a validation set to select hyper-parameters and other training choices.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
719 \{SDA,MLP\}0 are trained on NIST,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
720 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
721 The human error rate on digits is a lower bound because it does not count digits that were
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
722 recognized as letters. For comparison, the results found in the literature
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
723 on NIST digits classification using the same test set are included.}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
724 \label{tab:sda-vs-mlp-vs-humans}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
725 \begin{center}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
726 \begin{tabular}{|l|r|r|r|r|} \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
727 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
728 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
729 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
730 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
731 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
732 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
733 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
734 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
735 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
736 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
737 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
738 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
739 \end{tabular}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
740 \end{center}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
741 \end{table}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
742
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
743 \begin{table}[ht]
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
744 \caption{Relative change in error rates due to the use of perturbed training data,
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
745 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
746 A positive value indicates that training on the perturbed data helped for the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
747 given test set (the first 3 columns on the 62-class tasks and the last one is
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
748 on the clean 10-class digits). Clearly, the deep learning models did benefit more
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
749 from perturbed training data, even when testing on clean data, whereas the MLP
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
750 trained on perturbed data performed worse on the clean digits and about the same
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
751 on the clean characters. }
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
752 \label{tab:perturbation-effect}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
753 \begin{center}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
754 \begin{tabular}{|l|r|r|r|r|} \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
755 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
756 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
757 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
758 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
759 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
760 \end{tabular}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
761 \end{center}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
762 \end{table}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
763
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
764 \begin{table}[ht]
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
765 \caption{Test error rates and relative change in error rates due to the use of
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
766 a multi-task setting, i.e., training on each task in isolation vs training
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
767 for all three tasks together, for MLPs vs SDAs. The SDA benefits much
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
768 more from the multi-task setting. All experiments on only on the
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
769 unperturbed NIST data, using validation error for model selection.
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
770 Relative improvement is 1 - single-task error / multi-task error.}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
771 \label{tab:multi-task}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
772 \begin{center}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
773 \begin{tabular}{|l|r|r|r|} \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
774 & single-task & multi-task & relative \\
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
775 & setting & setting & improvement \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
776 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
777 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
778 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
779 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
780 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
781 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
782 \end{tabular}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
783 \end{center}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
784 \end{table}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
785
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
786 \fi
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
787
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
788 %\afterpage{\clearpage}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
789 %\clearpage
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
790 {
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
791 %\bibliographystyle{spbasic} % basic style, author-year citations
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
792 \bibliographystyle{plainnat}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
793 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
794 %\bibliographystyle{unsrtnat}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
795 %\bibliographystyle{apalike}
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
796 }
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
797
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
798
1f5d2d01b84d draft submission to AISTATS 2011
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
799 \end{document}