Mercurial > ift6266
annotate writeup/techreport.tex @ 612:21d53fd07f6e
reviews AISTATS
| author | Yoshua Bengio <bengioy@iro.umontreal.ca> |
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| date | Mon, 20 Dec 2010 11:54:35 -0500 |
| parents | 81c6fde68a8a |
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| rev | line source |
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| 582 | 1 \documentclass{article} % For LaTeX2e |
| 2 \usepackage{times} | |
| 3 \usepackage{wrapfig} | |
| 4 \usepackage{amsthm,amsmath,bbm} | |
| 5 \usepackage[psamsfonts]{amssymb} | |
| 6 \usepackage{algorithm,algorithmic} | |
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7 \usepackage[utf8]{inputenc} |
| 582 | 8 \usepackage{graphicx,subfigure} |
| 9 \usepackage[numbers]{natbib} | |
| 10 | |
| 11 \addtolength{\textwidth}{10mm} | |
| 12 \addtolength{\evensidemargin}{-5mm} | |
| 13 \addtolength{\oddsidemargin}{-5mm} | |
| 14 | |
| 15 %\setlength\parindent{0mm} | |
| 16 | |
| 17 \title{Deep Self-Taught Learning for Handwritten Character Recognition} | |
| 18 \author{ | |
| 19 Frédéric Bastien \and | |
| 20 Yoshua Bengio \and | |
| 21 Arnaud Bergeron \and | |
| 22 Nicolas Boulanger-Lewandowski \and | |
| 23 Thomas Breuel \and | |
| 24 Youssouf Chherawala \and | |
| 25 Moustapha Cisse \and | |
| 26 Myriam Côté \and | |
| 27 Dumitru Erhan \and | |
| 28 Jeremy Eustache \and | |
| 29 Xavier Glorot \and | |
| 30 Xavier Muller \and | |
| 31 Sylvain Pannetier Lebeuf \and | |
| 32 Razvan Pascanu \and | |
| 33 Salah Rifai \and | |
| 34 Francois Savard \and | |
| 35 Guillaume Sicard | |
| 36 } | |
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37 \date{June 3, 2010, Technical Report 1353, Dept. IRO, U. Montreal} |
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38 |
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39 \begin{document} |
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40 |
| 582 | 41 %\makeanontitle |
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42 \maketitle |
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45 \begin{abstract} |
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46 Recent theoretical and empirical work in statistical machine learning has demonstrated the importance of learning algorithms for deep architectures, i.e., function classes obtained by composing multiple non-linear transformations. Self-taught learning (exploiting unlabeled examples or examples from other distributions) has already been applied to deep learners, but mostly to show the advantage of unlabeled examples. Here we explore the advantage brought by {\em out-of-distribution examples}. For this purpose we developed a powerful generator of stochastic variations and noise processes for character images, including not only affine transformations but also slant, local elastic deformations, changes in thickness, background images, grey level changes, contrast, occlusion, and various types of noise. The out-of-distribution examples are obtained from these highly distorted images or by including examples of object classes different from those in the target test set. We show that {\em deep learners benefit more from out-of-distribution examples than a corresponding shallow learner}, at least in the area of handwritten character recognition. In fact, we show that they beat previously published results and reach human-level performance on both handwritten digit classification and 62-class handwritten character recognition. |
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47 \end{abstract} |
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49 |
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50 \section{Introduction} |
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52 |
| 582 | 53 {\bf Deep Learning} has emerged as a promising new area of research in |
| 54 statistical machine learning (see~\citet{Bengio-2009} for a review). | |
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55 Learning algorithms for deep architectures are centered on the learning |
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56 of useful representations of data, which are better suited to the task at hand, |
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57 and are organized in a hierarchy with multiple levels. |
| 582 | 58 This is in part inspired by observations of the mammalian visual cortex, |
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59 which consists of a chain of processing elements, each of which is associated with a |
| 582 | 60 different representation of the raw visual input. In fact, |
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61 it was found recently that the features learnt in deep architectures resemble |
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62 those observed in the first two of these stages (in areas V1 and V2 |
| 582 | 63 of visual cortex)~\citep{HonglakL2008}, and that they become more and |
| 64 more invariant to factors of variation (such as camera movement) in | |
| 65 higher layers~\citep{Goodfellow2009}. | |
| 66 Learning a hierarchy of features increases the | |
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67 ease and practicality of developing representations that are at once |
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68 tailored to specific tasks, yet are able to borrow statistical strength |
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69 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the |
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70 feature representation can lead to higher-level (more abstract, more |
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71 general) features that are more robust to unanticipated sources of |
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72 variance extant in real data. |
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73 |
| 582 | 74 {\bf Self-taught learning}~\citep{RainaR2007} is a paradigm that combines principles |
| 75 of semi-supervised and multi-task learning: the learner can exploit examples | |
| 76 that are unlabeled and possibly come from a distribution different from the target | |
| 77 distribution, e.g., from other classes than those of interest. | |
| 78 It has already been shown that deep learners can clearly take advantage of | |
| 79 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small}, | |
| 80 but more needs to be done to explore the impact | |
| 81 of {\em out-of-distribution} examples and of the multi-task setting | |
| 82 (one exception is~\citep{CollobertR2008}, which uses a different kind | |
| 83 of learning algorithm). In particular the {\em relative | |
| 84 advantage} of deep learning for these settings has not been evaluated. | |
| 85 The hypothesis discussed in the conclusion is that a deep hierarchy of features | |
| 86 may be better able to provide sharing of statistical strength | |
| 87 between different regions in input space or different tasks. | |
| 88 | |
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89 Whereas a deep architecture can in principle be more powerful than a |
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90 shallow one in terms of representation, depth appears to render the |
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91 training problem more difficult in terms of optimization and local minima. |
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92 It is also only recently that successful algorithms were proposed to |
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93 overcome some of these difficulties. All are based on unsupervised |
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94 learning, often in an greedy layer-wise ``unsupervised pre-training'' |
| 582 | 95 stage~\citep{Bengio-2009}. One of these layer initialization techniques, |
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96 applied here, is the Denoising |
| 582 | 97 Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see Figure~\ref{fig:da}), |
| 98 which | |
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99 performed similarly or better than previously proposed Restricted Boltzmann |
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100 Machines in terms of unsupervised extraction of a hierarchy of features |
| 582 | 101 useful for classification. Each layer is trained to denoise its |
| 102 input, creating a layer of features that can be used as input for the next layer. | |
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103 |
| 582 | 104 %The principle is that each layer starting from |
| 105 %the bottom is trained to encode its input (the output of the previous | |
| 106 %layer) and to reconstruct it from a corrupted version. After this | |
| 107 %unsupervised initialization, the stack of DAs can be | |
| 108 %converted into a deep supervised feedforward neural network and fine-tuned by | |
| 109 %stochastic gradient descent. | |
| 110 | |
| 111 % | |
| 112 In this paper we ask the following questions: | |
| 113 | |
| 114 %\begin{enumerate} | |
| 115 $\bullet$ %\item | |
| 116 Do the good results previously obtained with deep architectures on the | |
| 117 MNIST digit images generalize to the setting of a much larger and richer (but similar) | |
| 118 dataset, the NIST special database 19, with 62 classes and around 800k examples? | |
| 119 | |
| 120 $\bullet$ %\item | |
| 121 To what extent does the perturbation of input images (e.g. adding | |
| 122 noise, affine transformations, background images) make the resulting | |
| 123 classifiers better not only on similarly perturbed images but also on | |
| 124 the {\em original clean examples}? We study this question in the | |
| 125 context of the 62-class and 10-class tasks of the NIST special database 19. | |
| 126 | |
| 127 $\bullet$ %\item | |
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128 Do deep architectures {\em benefit {\bf more} from such out-of-distribution} |
| 582 | 129 examples, i.e. do they benefit more from the self-taught learning~\citep{RainaR2007} framework? |
| 130 We use highly perturbed examples to generate out-of-distribution examples. | |
| 131 | |
| 132 $\bullet$ %\item | |
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133 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more} |
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134 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than |
| 582 | 135 a corresponding shallow and purely supervised architecture? |
| 136 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case) | |
| 137 to answer this question. | |
| 138 %\end{enumerate} | |
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139 |
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140 Our experimental results provide positive evidence towards all of these questions, |
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141 as well as classifiers that reach human-level performance on 62-class isolated character |
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142 recognition and beat previously published results on the NIST dataset (special database 19). |
| 582 | 143 To achieve these results, we introduce in the next section a sophisticated system |
| 144 for stochastically transforming character images and then explain the methodology, | |
| 145 which is based on training with or without these transformed images and testing on | |
| 146 clean ones. We measure the relative advantage of out-of-distribution examples | |
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147 (perturbed or out-of-class) |
| 582 | 148 for a deep learner vs a supervised shallow one. |
| 149 Code for generating these transformations as well as for the deep learning | |
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150 algorithms are made available at {\tt http://hg.assembla.com/ift6266}. |
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151 We estimate the relative advantage for deep learners of training with |
| 582 | 152 other classes than those of interest, by comparing learners trained with |
| 153 62 classes with learners trained with only a subset (on which they | |
| 154 are then tested). | |
| 155 The conclusion discusses | |
| 156 the more general question of why deep learners may benefit so much from | |
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157 the self-taught learning framework. Since out-of-distribution data |
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158 (perturbed or from other related classes) is very common, this conclusion |
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159 is of practical importance. |
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160 |
| 582 | 161 %\vspace*{-3mm} |
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162 %\newpage |
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163 \section{Perturbed and Transformed Character Images} |
| 582 | 164 \label{s:perturbations} |
| 165 %\vspace*{-2mm} | |
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166 |
| 582 | 167 \begin{wrapfigure}[8]{l}{0.15\textwidth} |
| 168 %\begin{minipage}[b]{0.14\linewidth} | |
| 169 %\vspace*{-5mm} | |
| 170 \begin{center} | |
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171 \includegraphics[scale=.4]{Original.png}\\ |
| 582 | 172 {\bf Original} |
| 173 \end{center} | |
| 174 \end{wrapfigure} | |
| 175 %%\vspace{0.7cm} | |
| 176 %\end{minipage}% | |
| 177 %\hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 178 This section describes the different transformations we used to stochastically | |
| 179 transform $32 \times 32$ source images (such as the one on the left) | |
| 180 in order to obtain data from a larger distribution which | |
| 181 covers a domain substantially larger than the clean characters distribution from | |
| 182 which we start. | |
| 183 Although character transformations have been used before to | |
| 184 improve character recognizers, this effort is on a large scale both | |
| 185 in number of classes and in the complexity of the transformations, hence | |
| 186 in the complexity of the learning task. | |
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187 The code for these transformations (mostly python) is available at |
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188 {\tt http://hg.assembla.com/ift6266}. All the modules in the pipeline share |
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189 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the |
| 582 | 190 amount of deformation or noise introduced. |
| 191 There are two main parts in the pipeline. The first one, | |
| 192 from slant to pinch below, performs transformations. The second | |
| 193 part, from blur to contrast, adds different kinds of noise. | |
| 194 %\end{minipage} | |
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195 |
| 582 | 196 %\vspace*{1mm} |
| 197 \subsection{Transformations} | |
| 198 %{\large\bf 2.1 Transformations} | |
| 199 %\vspace*{1mm} | |
| 200 | |
| 201 \subsubsection*{Thickness} | |
| 541 | 202 |
| 582 | 203 %\begin{wrapfigure}[7]{l}{0.15\textwidth} |
| 204 \begin{minipage}[b]{0.14\linewidth} | |
| 205 %\centering | |
| 206 \begin{center} | |
| 207 \vspace*{-5mm} | |
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208 \includegraphics[scale=.4]{Thick_only.png}\\ |
| 582 | 209 %{\bf Thickness} |
| 210 \end{center} | |
| 211 \vspace{.6cm} | |
| 212 \end{minipage}% | |
| 213 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 214 %\end{wrapfigure} | |
| 215 To change character {\bf thickness}, morphological operators of dilation and erosion~\citep{Haralick87,Serra82} | |
| 541 | 216 are applied. The neighborhood of each pixel is multiplied |
| 217 element-wise with a {\em structuring element} matrix. | |
| 218 The pixel value is replaced by the maximum or the minimum of the resulting | |
| 219 matrix, respectively for dilation or erosion. Ten different structural elements with | |
| 220 increasing dimensions (largest is $5\times5$) were used. For each image, | |
| 221 randomly sample the operator type (dilation or erosion) with equal probability and one structural | |
| 582 | 222 element from a subset of the $n=round(m \times complexity)$ smallest structuring elements |
| 223 where $m=10$ for dilation and $m=6$ for erosion (to avoid completely erasing thin characters). | |
| 224 A neutral element (no transformation) | |
| 225 is always present in the set. | |
| 226 %%\vspace{.4cm} | |
| 227 \end{minipage} | |
| 541 | 228 |
| 582 | 229 \vspace{2mm} |
| 541 | 230 |
| 582 | 231 \subsubsection*{Slant} |
| 232 \vspace*{2mm} | |
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233 |
| 582 | 234 \begin{minipage}[b]{0.14\linewidth} |
| 235 \centering | |
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236 \includegraphics[scale=.4]{Slant_only.png}\\ |
| 582 | 237 %{\bf Slant} |
| 238 \end{minipage}% | |
| 239 \hspace{0.3cm} | |
| 240 \begin{minipage}[b]{0.83\linewidth} | |
| 241 %\centering | |
| 242 To produce {\bf slant}, each row of the image is shifted | |
| 243 proportionally to its height: $shift = round(slant \times height)$. | |
| 244 $slant \sim U[-complexity,complexity]$. | |
| 245 The shift is randomly chosen to be either to the left or to the right. | |
| 246 \vspace{5mm} | |
| 247 \end{minipage} | |
| 248 %\vspace*{-4mm} | |
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249 |
| 582 | 250 %\newpage |
| 541 | 251 |
| 582 | 252 \subsubsection*{Affine Transformations} |
| 541 | 253 |
| 582 | 254 \begin{minipage}[b]{0.14\linewidth} |
| 255 %\centering | |
| 256 %\begin{wrapfigure}[8]{l}{0.15\textwidth} | |
| 257 \begin{center} | |
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258 \includegraphics[scale=.4]{Affine_only.png} |
| 582 | 259 \vspace*{6mm} |
| 260 %{\small {\bf Affine \mbox{Transformation}}} | |
| 261 \end{center} | |
| 262 %\end{wrapfigure} | |
| 263 \end{minipage}% | |
| 264 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 265 \noindent A $2 \times 3$ {\bf affine transform} matrix (with | |
| 266 parameters $(a,b,c,d,e,f)$) is sampled according to the $complexity$. | |
| 267 Output pixel $(x,y)$ takes the value of input pixel | |
| 268 nearest to $(ax+by+c,dx+ey+f)$, | |
| 269 producing scaling, translation, rotation and shearing. | |
| 270 Marginal distributions of $(a,b,c,d,e,f)$ have been tuned to | |
| 271 forbid large rotations (to avoid confusing classes) but to give good | |
| 272 variability of the transformation: $a$ and $d$ $\sim U[1-3 | |
| 273 complexity,1+3\,complexity]$, $b$ and $e$ $\sim U[-3 \,complexity,3\, | |
| 274 complexity]$, and $c$ and $f \sim U[-4 \,complexity, 4 \, | |
| 275 complexity]$.\\ | |
| 276 \end{minipage} | |
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277 |
| 582 | 278 %\vspace*{-4.5mm} |
| 279 \subsubsection*{Local Elastic Deformations} | |
| 541 | 280 |
| 582 | 281 %\begin{minipage}[t]{\linewidth} |
| 282 %\begin{wrapfigure}[7]{l}{0.15\textwidth} | |
| 283 %\hspace*{-8mm} | |
| 284 \begin{minipage}[b]{0.14\linewidth} | |
| 285 %\centering | |
| 286 \begin{center} | |
| 287 \vspace*{5mm} | |
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288 \includegraphics[scale=.4]{Localelasticdistorsions_only.png} |
| 582 | 289 %{\bf Local Elastic Deformation} |
| 290 \end{center} | |
| 291 %\end{wrapfigure} | |
| 292 \end{minipage}% | |
| 293 \hspace{3mm} | |
| 294 \begin{minipage}[b]{0.85\linewidth} | |
| 295 %%\vspace*{-20mm} | |
| 296 The {\bf local elastic deformation} | |
| 297 module induces a ``wiggly'' effect in the image, following~\citet{SimardSP03-short}, | |
| 298 which provides more details. | |
| 299 The intensity of the displacement fields is given by | |
| 300 $\alpha = \sqrt[3]{complexity} \times 10.0$, which are | |
| 301 convolved with a Gaussian 2D kernel (resulting in a blur) of | |
| 302 standard deviation $\sigma = 10 - 7 \times\sqrt[3]{complexity}$. | |
| 303 \vspace{2mm} | |
| 304 \end{minipage} | |
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305 |
| 582 | 306 \vspace*{4mm} |
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307 |
| 582 | 308 \subsubsection*{Pinch} |
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309 |
| 582 | 310 \begin{minipage}[b]{0.14\linewidth} |
| 311 %\centering | |
| 312 %\begin{wrapfigure}[7]{l}{0.15\textwidth} | |
| 313 %\vspace*{-5mm} | |
| 314 \begin{center} | |
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315 \includegraphics[scale=.4]{Pinch_only.png}\\ |
| 582 | 316 \vspace*{15mm} |
| 317 %{\bf Pinch} | |
| 318 \end{center} | |
| 319 %\end{wrapfigure} | |
| 320 %%\vspace{.6cm} | |
| 321 \end{minipage}% | |
| 322 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 323 The {\bf pinch} module applies the ``Whirl and pinch'' GIMP filter with whirl set to 0. | |
| 324 A pinch is ``similar to projecting the image onto an elastic | |
| 541 | 325 surface and pressing or pulling on the center of the surface'' (GIMP documentation manual). |
| 582 | 326 For a square input image, draw a radius-$r$ disk |
| 327 around its center $C$. Any pixel $P$ belonging to | |
| 328 that disk has its value replaced by | |
| 329 the value of a ``source'' pixel in the original image, | |
| 330 on the line that goes through $C$ and $P$, but | |
| 331 at some other distance $d_2$. Define $d_1=distance(P,C)$ | |
| 332 and $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times | |
| 333 d_1$, where $pinch$ is a parameter of the filter. | |
| 541 | 334 The actual value is given by bilinear interpolation considering the pixels |
| 335 around the (non-integer) source position thus found. | |
| 336 Here $pinch \sim U[-complexity, 0.7 \times complexity]$. | |
| 582 | 337 %%\vspace{1.5cm} |
| 338 \end{minipage} | |
| 541 | 339 |
| 582 | 340 %\vspace{1mm} |
|
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341 |
| 582 | 342 %{\large\bf 2.2 Injecting Noise} |
| 343 \subsection{Injecting Noise} | |
| 344 %\vspace{2mm} | |
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345 |
| 582 | 346 \subsubsection*{Motion Blur} |
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347 |
| 582 | 348 %%\vspace*{-.2cm} |
| 349 \begin{minipage}[t]{0.14\linewidth} | |
| 350 \centering | |
| 351 \vspace*{0mm} | |
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352 \includegraphics[scale=.4]{Motionblur_only.png} |
| 582 | 353 %{\bf Motion Blur} |
| 354 \end{minipage}% | |
| 355 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 356 %%\vspace*{.5mm} | |
| 357 \vspace*{2mm} | |
| 358 The {\bf motion blur} module is GIMP's ``linear motion blur'', which | |
| 359 has parameters $length$ and $angle$. The value of | |
| 360 a pixel in the final image is approximately the mean of the first $length$ pixels | |
| 361 found by moving in the $angle$ direction, | |
| 362 $angle \sim U[0,360]$ degrees, and $length \sim {\rm Normal}(0,(3 \times complexity)^2)$. | |
| 363 %\vspace{5mm} | |
| 364 \end{minipage} | |
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365 |
| 582 | 366 %\vspace*{1mm} |
| 367 | |
| 368 \subsubsection*{Occlusion} | |
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369 |
| 582 | 370 \begin{minipage}[t]{0.14\linewidth} |
| 371 \centering | |
| 372 \vspace*{3mm} | |
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373 \includegraphics[scale=.4]{occlusion_only.png}\\ |
| 582 | 374 %{\bf Occlusion} |
| 375 %%\vspace{.5cm} | |
| 376 \end{minipage}% | |
| 377 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 378 %\vspace*{-18mm} | |
| 379 The {\bf occlusion} module selects a random rectangle from an {\em occluder} character | |
| 380 image and places it over the original {\em occluded} | |
| 381 image. Pixels are combined by taking the max(occluder, occluded), | |
| 382 i.e. keeping the lighter ones. | |
| 383 The rectangle corners | |
| 541 | 384 are sampled so that larger complexity gives larger rectangles. |
| 385 The destination position in the occluded image are also sampled | |
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386 according to a normal distribution. |
| 582 | 387 This module is skipped with probability 60\%. |
| 388 %%\vspace{7mm} | |
| 389 \end{minipage} | |
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390 |
| 582 | 391 %\vspace*{1mm} |
| 392 \subsubsection*{Gaussian Smoothing} | |
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393 |
| 582 | 394 %\begin{wrapfigure}[8]{l}{0.15\textwidth} |
| 395 %\vspace*{-6mm} | |
| 396 \begin{minipage}[t]{0.14\linewidth} | |
| 397 \begin{center} | |
| 398 %\centering | |
| 399 \vspace*{6mm} | |
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400 \includegraphics[scale=.4]{Bruitgauss_only.png} |
| 582 | 401 %{\bf Gaussian Smoothing} |
| 402 \end{center} | |
| 403 %\end{wrapfigure} | |
| 404 %%\vspace{.5cm} | |
| 405 \end{minipage}% | |
| 406 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth} | |
| 407 With the {\bf Gaussian smoothing} module, | |
| 408 different regions of the image are spatially smoothed. | |
| 409 This is achieved by first convolving | |
| 410 the image with an isotropic Gaussian kernel of | |
| 541 | 411 size and variance chosen uniformly in the ranges $[12,12 + 20 \times |
| 582 | 412 complexity]$ and $[2,2 + 6 \times complexity]$. This filtered image is normalized |
| 413 between $0$ and $1$. We also create an isotropic weighted averaging window, of the | |
| 541 | 414 kernel size, with maximum value at the center. For each image we sample |
| 415 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be | |
| 416 averaging centers between the original image and the filtered one. We | |
| 417 initialize to zero a mask matrix of the image size. For each selected pixel | |
| 582 | 418 we add to the mask the averaging window centered on it. The final image is |
| 419 computed from the following element-wise operation: $\frac{image + filtered\_image | |
| 420 \times mask}{mask+1}$. | |
| 421 This module is skipped with probability 75\%. | |
| 422 \end{minipage} | |
| 423 | |
| 424 %\newpage | |
| 425 | |
| 426 %\vspace*{-9mm} | |
| 427 \subsubsection*{Permute Pixels} | |
| 541 | 428 |
| 582 | 429 %\hspace*{-3mm}\begin{minipage}[t]{0.18\linewidth} |
| 430 %\centering | |
| 431 \begin{minipage}[t]{0.14\textwidth} | |
| 432 %\begin{wrapfigure}[7]{l}{ | |
| 433 %\vspace*{-5mm} | |
| 434 \begin{center} | |
| 435 \vspace*{1mm} | |
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436 \includegraphics[scale=.4]{Permutpixel_only.png} |
| 582 | 437 %{\small\bf Permute Pixels} |
| 438 \end{center} | |
| 439 %\end{wrapfigure} | |
| 440 \end{minipage}% | |
| 441 \hspace{3mm}\begin{minipage}[t]{0.86\linewidth} | |
| 442 \vspace*{1mm} | |
| 443 %%\vspace*{-20mm} | |
| 444 This module {\bf permutes neighbouring pixels}. It first selects a | |
| 445 fraction $\frac{complexity}{3}$ of pixels randomly in the image. Each | |
| 446 of these pixels is then sequentially exchanged with a random pixel | |
| 447 among its four nearest neighbors (on its left, right, top or bottom). | |
| 448 This module is skipped with probability 80\%.\\ | |
| 449 %\vspace*{1mm} | |
| 450 \end{minipage} | |
| 451 | |
| 452 %\vspace{-3mm} | |
| 453 | |
| 454 \subsubsection*{Gaussian Noise} | |
| 455 | |
| 456 \begin{minipage}[t]{0.14\textwidth} | |
| 457 %\begin{wrapfigure}[7]{l}{ | |
| 458 %%\vspace*{-3mm} | |
| 459 \begin{center} | |
| 460 %\hspace*{-3mm}\begin{minipage}[t]{0.18\linewidth} | |
| 461 %\centering | |
| 462 \vspace*{0mm} | |
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463 \includegraphics[scale=.4]{Distorsiongauss_only.png} |
| 582 | 464 %{\small \bf Gauss. Noise} |
| 465 \end{center} | |
| 466 %\end{wrapfigure} | |
| 467 \end{minipage}% | |
| 468 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth} | |
| 469 \vspace*{1mm} | |
| 470 %\vspace*{12mm} | |
| 471 The {\bf Gaussian noise} module simply adds, to each pixel of the image independently, a | |
| 472 noise $\sim Normal(0,(\frac{complexity}{10})^2)$. | |
| 473 This module is skipped with probability 70\%. | |
| 474 %%\vspace{1.1cm} | |
| 475 \end{minipage} | |
| 541 | 476 |
| 582 | 477 %\vspace*{1.2cm} |
| 478 | |
| 479 \subsubsection*{Background Image Addition} | |
| 480 | |
| 481 \begin{minipage}[t]{\linewidth} | |
| 482 \begin{minipage}[t]{0.14\linewidth} | |
| 483 \centering | |
| 484 \vspace*{0mm} | |
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485 \includegraphics[scale=.4]{background_other_only.png} |
| 582 | 486 %{\small \bf Bg Image} |
| 487 \end{minipage}% | |
| 488 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 489 \vspace*{1mm} | |
| 490 Following~\citet{Larochelle-jmlr-2009}, the {\bf background image} module adds a random | |
| 491 background image behind the letter, from a randomly chosen natural image, | |
| 492 with contrast adjustments depending on $complexity$, to preserve | |
| 493 more or less of the original character image. | |
| 494 %%\vspace{.8cm} | |
| 495 \end{minipage} | |
| 496 \end{minipage} | |
| 497 %%\vspace{-.7cm} | |
| 498 | |
| 499 \subsubsection*{Salt and Pepper Noise} | |
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500 |
| 582 | 501 \begin{minipage}[t]{0.14\linewidth} |
| 502 \centering | |
| 503 \vspace*{0mm} | |
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504 \includegraphics[scale=.4]{Poivresel_only.png} |
| 582 | 505 %{\small \bf Salt \& Pepper} |
| 506 \end{minipage}% | |
| 507 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 508 \vspace*{1mm} | |
| 509 The {\bf salt and pepper noise} module adds noise $\sim U[0,1]$ to random subsets of pixels. | |
| 510 The number of selected pixels is $0.2 \times complexity$. | |
| 511 This module is skipped with probability 75\%. | |
| 512 %%\vspace{.9cm} | |
| 513 \end{minipage} | |
| 514 %%\vspace{-.7cm} | |
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515 |
| 582 | 516 %\vspace{1mm} |
| 517 \subsubsection*{Scratches} | |
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518 |
| 582 | 519 \begin{minipage}[t]{0.14\textwidth} |
| 520 %\begin{wrapfigure}[7]{l}{ | |
| 521 %\begin{minipage}[t]{0.14\linewidth} | |
| 522 %\centering | |
| 523 \begin{center} | |
| 524 \vspace*{4mm} | |
| 525 %\hspace*{-1mm} | |
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526 \includegraphics[scale=.4]{Rature_only.png}\\ |
| 582 | 527 %{\bf Scratches} |
| 528 \end{center} | |
| 529 \end{minipage}% | |
| 530 %\end{wrapfigure} | |
| 531 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth} | |
| 532 %%\vspace{.4cm} | |
| 533 The {\bf scratches} module places line-like white patches on the image. The | |
| 541 | 534 lines are heavily transformed images of the digit ``1'' (one), chosen |
| 582 | 535 at random among 500 such 1 images, |
| 541 | 536 randomly cropped and rotated by an angle $\sim Normal(0,(100 \times |
| 582 | 537 complexity)^2$ (in degrees), using bi-cubic interpolation. |
| 541 | 538 Two passes of a grey-scale morphological erosion filter |
| 539 are applied, reducing the width of the line | |
| 540 by an amount controlled by $complexity$. | |
| 582 | 541 This module is skipped with probability 85\%. The probabilities |
| 542 of applying 1, 2, or 3 patches are (50\%,30\%,20\%). | |
| 543 \end{minipage} | |
| 428 | 544 |
| 582 | 545 %\vspace*{1mm} |
| 428 | 546 |
| 582 | 547 \subsubsection*{Grey Level and Contrast Changes} |
| 428 | 548 |
| 582 | 549 \begin{minipage}[t]{0.15\linewidth} |
| 550 \centering | |
| 551 \vspace*{0mm} | |
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552 \includegraphics[scale=.4]{Contrast_only.png} |
| 582 | 553 %{\bf Grey Level \& Contrast} |
| 554 \end{minipage}% | |
| 555 \hspace{3mm}\begin{minipage}[t]{0.85\linewidth} | |
| 556 \vspace*{1mm} | |
| 557 The {\bf grey level and contrast} module changes the contrast by changing grey levels, and may invert the image polarity (white | |
| 558 to black and black to white). The contrast is $C \sim U[1-0.85 \times complexity,1]$ | |
| 559 so the image is normalized into $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The | |
| 560 polarity is inverted with probability 50\%. | |
| 561 %%\vspace{.7cm} | |
| 562 \end{minipage} | |
| 563 %\vspace{2mm} | |
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564 |
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565 |
| 582 | 566 \iffalse |
| 567 \begin{figure}[ht] | |
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568 \centerline{\resizebox{.9\textwidth}{!}{\includegraphics{example_t.png}}}\\ |
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569 \caption{Illustration of the pipeline of stochastic |
| 582 | 570 transformations applied to the image of a lower-case \emph{t} |
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571 (the upper left image). Each image in the pipeline (going from |
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572 left to right, first top line, then bottom line) shows the result |
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573 of applying one of the modules in the pipeline. The last image |
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574 (bottom right) is used as training example.} |
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575 \label{fig:pipeline} |
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576 \end{figure} |
| 582 | 577 \fi |
| 578 | |
| 579 %\vspace*{-3mm} | |
| 580 \section{Experimental Setup} | |
| 581 %\vspace*{-1mm} | |
| 582 | |
| 583 Much previous work on deep learning had been performed on | |
| 584 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009}, | |
| 585 with 60~000 examples, and variants involving 10~000 | |
| 586 examples~\citep{Larochelle-jmlr-toappear-2008,VincentPLarochelleH2008}. | |
| 587 The focus here is on much larger training sets, from 10 times to | |
| 588 to 1000 times larger, and 62 classes. | |
| 589 | |
| 590 The first step in constructing the larger datasets (called NISTP and P07) is to sample from | |
| 591 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas}, | |
| 592 and {\bf OCR data} (scanned machine printed characters). Once a character | |
| 593 is sampled from one of these sources (chosen randomly), the second step is to | |
| 594 apply a pipeline of transformations and/or noise processes described in section \ref{s:perturbations}. | |
| 595 | |
| 596 To provide a baseline of error rate comparison we also estimate human performance | |
| 597 on both the 62-class task and the 10-class digits task. | |
| 598 We compare the best Multi-Layer Perceptrons (MLP) against | |
| 599 the best Stacked Denoising Auto-encoders (SDA), when | |
| 600 both models' hyper-parameters are selected to minimize the validation set error. | |
| 601 We also provide a comparison against a precise estimate | |
| 602 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 603 service (http://mturk.com). | |
| 604 AMT users are paid small amounts | |
| 605 of money to perform tasks for which human intelligence is required. | |
| 606 Mechanical Turk has been used extensively in natural language processing and vision. | |
| 607 %processing \citep{SnowEtAl2008} and vision | |
| 608 %\citep{SorokinAndForsyth2008,whitehill09}. | |
| 609 AMT users were presented | |
| 610 with 10 character images (from a test set) and asked to choose 10 corresponding ASCII | |
| 611 characters. They were forced to choose a single character class (either among the | |
| 612 62 or 10 character classes) for each image. | |
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613 80 subjects classified 2500 images per (dataset,task) pair. |
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614 Different humans labelers sometimes provided a different label for the same |
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615 example, and we were able to estimate the error variance due to this effect |
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616 because each image was classified by 3 different persons. |
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617 The average error of humans on the 62-class task NIST test set |
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618 is 18.2\%, with a standard error of 0.1\%. |
| 582 | 619 |
| 620 %\vspace*{-3mm} | |
| 621 \subsection{Data Sources} | |
| 622 %\vspace*{-2mm} | |
| 623 | |
| 624 %\begin{itemize} | |
| 625 %\item | |
| 626 {\bf NIST.} | |
| 627 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995}, | |
| 628 widely used for training and testing character | |
| 629 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}. | |
| 630 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications, | |
| 631 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes | |
| 632 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity. | |
| 633 The fourth partition (called $hsf_4$, 82587 examples), | |
| 634 experimentally recognized to be the most difficult one, is the one recommended | |
| 635 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} | |
| 636 for that purpose. We randomly split the remainder (731668 examples) into a training set and a validation set for | |
| 637 model selection. | |
| 638 The performances reported by previous work on that dataset mostly use only the digits. | |
| 639 Here we use all the classes both in the training and testing phase. This is especially | |
| 640 useful to estimate the effect of a multi-task setting. | |
| 641 The distribution of the classes in the NIST training and test sets differs | |
| 642 substantially, with relatively many more digits in the test set, and a more uniform distribution | |
| 643 of letters in the test set (whereas in the training set they are distributed | |
| 644 more like in natural text). | |
| 645 %\vspace*{-1mm} | |
| 646 | |
| 647 %\item | |
| 648 {\bf Fonts.} | |
| 649 In order to have a good variety of sources we downloaded an important number of free fonts from: | |
| 650 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}. | |
| 651 % TODO: pointless to anonymize, it's not pointing to our work | |
| 652 Including the operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from. | |
| 653 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, | |
| 654 directly as input to our models. | |
| 655 %\vspace*{-1mm} | |
| 656 | |
| 657 %\item | |
| 658 {\bf Captchas.} | |
| 659 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for | |
| 660 generating characters of the same format as the NIST dataset. This software is based on | |
| 661 a random character class generator and various kinds of transformations similar to those described in the previous sections. | |
| 662 In order to increase the variability of the data generated, many different fonts are used for generating the characters. | |
| 663 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity | |
| 664 depending on the value of the complexity parameter provided by the user of the data source. | |
| 665 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class? | |
| 666 %\vspace*{-1mm} | |
| 667 | |
| 668 %\item | |
| 669 {\bf OCR data.} | |
| 670 A large set (2 million) of scanned, OCRed and manually verified machine-printed | |
| 671 characters where included as an | |
| 672 additional source. This set is part of a larger corpus being collected by the Image Understanding | |
| 673 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern | |
| 674 ({\tt http://www.iupr.com}), and which will be publicly released. | |
| 675 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this | |
| 676 %\end{itemize} | |
| 677 | |
| 678 %\vspace*{-3mm} | |
| 679 \subsection{Data Sets} | |
| 680 %\vspace*{-2mm} | |
| 681 | |
| 682 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label | |
| 683 from one of the 62 character classes. | |
| 684 %\begin{itemize} | |
| 685 %\vspace*{-1mm} | |
| 686 | |
| 687 %\item | |
| 688 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has | |
| 689 \{651668 / 80000 / 82587\} \{training / validation / test\} examples. | |
| 690 %\vspace*{-1mm} | |
| 691 | |
| 692 %\item | |
| 693 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources | |
| 694 and sending them through the transformation pipeline described in section \ref{s:perturbations}. | |
| 695 For each new example to generate, a data source is selected with probability $10\%$ from the fonts, | |
| 696 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the | |
| 697 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$. | |
| 698 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples. | |
| 699 %\vspace*{-1mm} | |
| 700 | |
| 701 %\item | |
| 702 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources) | |
| 703 except that we only apply | |
| 704 transformations from slant to pinch. Therefore, the character is | |
| 705 transformed but no additional noise is added to the image, giving images | |
| 706 closer to the NIST dataset. | |
| 707 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples. | |
| 708 %\end{itemize} | |
| 709 | |
| 710 %\vspace*{-3mm} | |
| 711 \subsection{Models and their Hyperparameters} | |
| 712 %\vspace*{-2mm} | |
| 713 | |
| 714 The experiments are performed using MLPs (with a single | |
| 715 hidden layer) and SDAs. | |
| 716 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.} | |
| 717 | |
| 718 {\bf Multi-Layer Perceptrons (MLP).} | |
| 719 Whereas previous work had compared deep architectures to both shallow MLPs and | |
| 720 SVMs, we only compared to MLPs here because of the very large datasets used | |
| 721 (making the use of SVMs computationally challenging because of their quadratic | |
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722 scaling behavior). Preliminary experiments on training SVMs (libSVM) with subsets of the training |
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723 set allowing the program to fit in memory yielded substantially worse results |
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724 than those obtained with MLPs. For training on nearly a billion examples |
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725 (with the perturbed data), the MLPs and SDA are much more convenient than |
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726 classifiers based on kernel methods. |
| 582 | 727 The MLP has a single hidden layer with $\tanh$ activation functions, and softmax (normalized |
| 728 exponentials) on the output layer for estimating $P(class | image)$. | |
| 729 The number of hidden units is taken in $\{300,500,800,1000,1500\}$. | |
| 730 Training examples are presented in minibatches of size 20. A constant learning | |
| 731 rate was chosen among $\{0.001, 0.01, 0.025, 0.075, 0.1, 0.5\}$. | |
| 732 %through preliminary experiments (measuring performance on a validation set), | |
| 733 %and $0.1$ (which was found to work best) was then selected for optimizing on | |
| 734 %the whole training sets. | |
| 735 %\vspace*{-1mm} | |
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736 |
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737 |
| 582 | 738 {\bf Stacked Denoising Auto-Encoders (SDA).} |
| 739 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs) | |
| 740 can be used to initialize the weights of each layer of a deep MLP (with many hidden | |
| 741 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006}, | |
| 742 apparently setting parameters in the | |
| 743 basin of attraction of supervised gradient descent yielding better | |
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744 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised |
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745 pre-training phase} uses all of the training images but not the training labels. |
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746 Each layer is trained in turn to produce a new representation of its input |
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747 (starting from the raw pixels). |
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748 It is hypothesized that the |
| 582 | 749 advantage brought by this procedure stems from a better prior, |
| 750 on the one hand taking advantage of the link between the input | |
| 751 distribution $P(x)$ and the conditional distribution of interest | |
| 752 $P(y|x)$ (like in semi-supervised learning), and on the other hand | |
| 753 taking advantage of the expressive power and bias implicit in the | |
| 754 deep architecture (whereby complex concepts are expressed as | |
| 755 compositions of simpler ones through a deep hierarchy). | |
| 756 | |
| 757 \begin{figure}[ht] | |
| 758 %\vspace*{-2mm} | |
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759 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{denoising_autoencoder_small.pdf}}} |
| 582 | 760 %\vspace*{-2mm} |
| 761 \caption{Illustration of the computations and training criterion for the denoising | |
| 762 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of | |
| 763 the layer (i.e. raw input or output of previous layer) | |
| 764 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$. | |
| 765 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which | |
| 766 is compared to the uncorrupted input $x$ through the loss function | |
| 767 $L_H(x,z)$, whose expected value is approximately minimized during training | |
| 768 by tuning $\theta$ and $\theta'$.} | |
| 769 \label{fig:da} | |
| 770 %\vspace*{-2mm} | |
| 771 \end{figure} | |
| 772 | |
| 773 Here we chose to use the Denoising | |
| 774 Auto-encoder~\citep{VincentPLarochelleH2008} as the building block for | |
| 775 these deep hierarchies of features, as it is simple to train and | |
| 776 explain (see Figure~\ref{fig:da}, as well as | |
| 777 tutorial and code there: {\tt http://deeplearning.net/tutorial}), | |
| 778 provides efficient inference, and yielded results | |
| 779 comparable or better than RBMs in series of experiments | |
| 780 \citep{VincentPLarochelleH2008}. During training, a Denoising | |
| 781 Auto-encoder is presented with a stochastically corrupted version | |
| 782 of the input and trained to reconstruct the uncorrupted input, | |
| 783 forcing the hidden units to represent the leading regularities in | |
| 784 the data. Here we use the random binary masking corruption | |
| 785 (which sets to 0 a random subset of the inputs). | |
| 786 Once it is trained, in a purely unsupervised way, | |
| 787 its hidden units' activations can | |
| 788 be used as inputs for training a second one, etc. | |
| 789 After this unsupervised pre-training stage, the parameters | |
| 790 are used to initialize a deep MLP, which is fine-tuned by | |
| 791 the same standard procedure used to train them (see previous section). | |
| 792 The SDA hyper-parameters are the same as for the MLP, with the addition of the | |
| 793 amount of corruption noise (we used the masking noise process, whereby a | |
| 794 fixed proportion of the input values, randomly selected, are zeroed), and a | |
| 795 separate learning rate for the unsupervised pre-training stage (selected | |
| 796 from the same above set). The fraction of inputs corrupted was selected | |
| 797 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number | |
| 798 of hidden layers but it was fixed to 3 based on previous work with | |
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799 SDAs on MNIST~\citep{VincentPLarochelleH2008}. The size of the hidden |
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800 layers was kept constant across hidden layers, and the best results |
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801 were obtained with the largest values that we could experiment |
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802 with given our patience, with 1000 hidden units. |
| 582 | 803 |
| 804 %\vspace*{-1mm} | |
| 805 | |
| 806 \begin{figure}[ht] | |
| 807 %\vspace*{-2mm} | |
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808 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{error_rates_charts.pdf}}} |
| 582 | 809 %\vspace*{-3mm} |
| 810 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained | |
| 811 on NIST, 1 on NISTP, and 2 on P07. Left: overall results | |
| 812 of all models, on NIST and NISTP test sets. | |
| 813 Right: error rates on NIST test digits only, along with the previous results from | |
| 814 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005} | |
| 815 respectively based on ART, nearest neighbors, MLPs, and SVMs.} | |
| 816 \label{fig:error-rates-charts} | |
| 817 %\vspace*{-2mm} | |
|
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818 \end{figure} |
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819 |
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820 |
| 582 | 821 \begin{figure}[ht] |
| 822 %\vspace*{-3mm} | |
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823 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{improvements_charts.pdf}}} |
| 582 | 824 %\vspace*{-3mm} |
| 825 \caption{Relative improvement in error rate due to self-taught learning. | |
| 826 Left: Improvement (or loss, when negative) | |
| 827 induced by out-of-distribution examples (perturbed data). | |
| 828 Right: Improvement (or loss, when negative) induced by multi-task | |
| 829 learning (training on all classes and testing only on either digits, | |
| 830 upper case, or lower-case). The deep learner (SDA) benefits more from | |
| 831 both self-taught learning scenarios, compared to the shallow MLP.} | |
| 832 \label{fig:improvements-charts} | |
| 833 %\vspace*{-2mm} | |
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834 \end{figure} |
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835 |
| 582 | 836 \section{Experimental Results} |
| 837 %\vspace*{-2mm} | |
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838 |
| 582 | 839 %%\vspace*{-1mm} |
| 840 %\subsection{SDA vs MLP vs Humans} | |
| 841 %%\vspace*{-1mm} | |
| 842 The models are either trained on NIST (MLP0 and SDA0), | |
| 843 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested | |
| 844 on either NIST, NISTP or P07, either on the 62-class task | |
| 845 or on the 10-digits task. Training (including about half | |
| 846 for unsupervised pre-training, for DAs) on the larger | |
| 847 datasets takes around one day on a GPU-285. | |
| 848 Figure~\ref{fig:error-rates-charts} summarizes the results obtained, | |
| 849 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1, | |
| 850 SDA2), along with the previous results on the digits NIST special database | |
| 851 19 test set from the literature, respectively based on ARTMAP neural | |
| 852 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search | |
| 853 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs | |
| 854 ~\citep{Milgram+al-2005}. More detailed and complete numerical results | |
| 855 (figures and tables, including standard errors on the error rates) can be | |
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856 found in Appendix. |
| 582 | 857 The deep learner not only outperformed the shallow ones and |
| 858 previously published performance (in a statistically and qualitatively | |
| 859 significant way) but when trained with perturbed data | |
| 860 reaches human performance on both the 62-class task | |
| 861 and the 10-class (digits) task. | |
| 862 17\% error (SDA1) or 18\% error (humans) may seem large but a large | |
| 863 majority of the errors from humans and from SDA1 are from out-of-context | |
| 864 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a | |
| 865 ``c'' and a ``C'' are often indistinguishible). | |
| 438 | 866 |
| 582 | 867 In addition, as shown in the left of |
| 868 Figure~\ref{fig:improvements-charts}, the relative improvement in error | |
| 869 rate brought by self-taught learning is greater for the SDA, and these | |
| 870 differences with the MLP are statistically and qualitatively | |
| 871 significant. | |
| 872 The left side of the figure shows the improvement to the clean | |
| 873 NIST test set error brought by the use of out-of-distribution examples | |
| 874 (i.e. the perturbed examples examples from NISTP or P07). | |
| 875 Relative percent change is measured by taking | |
| 876 $100 \% \times$ (original model's error / perturbed-data model's error - 1). | |
| 877 The right side of | |
| 878 Figure~\ref{fig:improvements-charts} shows the relative improvement | |
| 879 brought by the use of a multi-task setting, in which the same model is | |
| 880 trained for more classes than the target classes of interest (i.e. training | |
| 881 with all 62 classes when the target classes are respectively the digits, | |
| 882 lower-case, or upper-case characters). Again, whereas the gain from the | |
| 883 multi-task setting is marginal or negative for the MLP, it is substantial | |
| 884 for the SDA. Note that to simplify these multi-task experiments, only the original | |
| 885 NIST dataset is used. For example, the MLP-digits bar shows the relative | |
| 886 percent improvement in MLP error rate on the NIST digits test set | |
| 887 is $100\% \times$ (single-task | |
| 888 model's error / multi-task model's error - 1). The single-task model is | |
| 889 trained with only 10 outputs (one per digit), seeing only digit examples, | |
| 890 whereas the multi-task model is trained with 62 outputs, with all 62 | |
| 891 character classes as examples. Hence the hidden units are shared across | |
| 892 all tasks. For the multi-task model, the digit error rate is measured by | |
| 893 comparing the correct digit class with the output class associated with the | |
| 894 maximum conditional probability among only the digit classes outputs. The | |
| 895 setting is similar for the other two target classes (lower case characters | |
| 896 and upper case characters). | |
| 897 %%\vspace*{-1mm} | |
| 898 %\subsection{Perturbed Training Data More Helpful for SDA} | |
| 899 %%\vspace*{-1mm} | |
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| 582 | 901 %%\vspace*{-1mm} |
| 902 %\subsection{Multi-Task Learning Effects} | |
| 903 %%\vspace*{-1mm} | |
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| 582 | 905 \iffalse |
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906 As previously seen, the SDA is better able to benefit from the |
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907 transformations applied to the data than the MLP. In this experiment we |
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908 define three tasks: recognizing digits (knowing that the input is a digit), |
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909 recognizing upper case characters (knowing that the input is one), and |
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910 recognizing lower case characters (knowing that the input is one). We |
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911 consider the digit classification task as the target task and we want to |
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912 evaluate whether training with the other tasks can help or hurt, and |
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913 whether the effect is different for MLPs versus SDAs. The goal is to find |
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914 out if deep learning can benefit more (or less) from multiple related tasks |
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915 (i.e. the multi-task setting) compared to a corresponding purely supervised |
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916 shallow learner. |
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917 |
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918 We use a single hidden layer MLP with 1000 hidden units, and a SDA |
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919 with 3 hidden layers (1000 hidden units per layer), pre-trained and |
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920 fine-tuned on NIST. |
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921 |
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922 Our results show that the MLP benefits marginally from the multi-task setting |
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923 in the case of digits (5\% relative improvement) but is actually hurt in the case |
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924 of characters (respectively 3\% and 4\% worse for lower and upper class characters). |
| 582 | 925 On the other hand the SDA benefited from the multi-task setting, with relative |
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926 error rate improvements of 27\%, 15\% and 13\% respectively for digits, |
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927 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. |
| 582 | 928 \fi |
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929 |
| 582 | 930 |
| 931 %\vspace*{-2mm} | |
| 932 \section{Conclusions and Discussion} | |
| 933 %\vspace*{-2mm} | |
| 934 | |
| 935 We have found that the self-taught learning framework is more beneficial | |
| 936 to a deep learner than to a traditional shallow and purely | |
| 937 supervised learner. More precisely, | |
| 938 the answers are positive for all the questions asked in the introduction. | |
| 939 %\begin{itemize} | |
| 940 | |
| 941 $\bullet$ %\item | |
| 942 {\bf Do the good results previously obtained with deep architectures on the | |
| 943 MNIST digits generalize to a much larger and richer (but similar) | |
| 944 dataset, the NIST special database 19, with 62 classes and around 800k examples}? | |
| 945 Yes, the SDA {\em systematically outperformed the MLP and all the previously | |
| 946 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level | |
|
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947 performance} at around 17\% error on the 62-class task and 1.4\% on the digits, |
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948 and beating previously published results on the same data. |
| 582 | 949 |
| 950 $\bullet$ %\item | |
| 951 {\bf To what extent do self-taught learning scenarios help deep learners, | |
| 952 and do they help them more than shallow supervised ones}? | |
| 953 We found that distorted training examples not only made the resulting | |
| 954 classifier better on similarly perturbed images but also on | |
| 955 the {\em original clean examples}, and more importantly and more novel, | |
| 956 that deep architectures benefit more from such {\em out-of-distribution} | |
| 957 examples. MLPs were helped by perturbed training examples when tested on perturbed input | |
| 958 images (65\% relative improvement on NISTP) | |
| 959 but only marginally helped (5\% relative improvement on all classes) | |
| 960 or even hurt (10\% relative loss on digits) | |
| 961 with respect to clean examples . On the other hand, the deep SDAs | |
| 962 were significantly boosted by these out-of-distribution examples. | |
| 963 Similarly, whereas the improvement due to the multi-task setting was marginal or | |
| 964 negative for the MLP (from +5.6\% to -3.6\% relative change), | |
| 965 it was quite significant for the SDA (from +13\% to +27\% relative change), | |
| 966 which may be explained by the arguments below. | |
| 967 %\end{itemize} | |
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968 |
| 582 | 969 In the original self-taught learning framework~\citep{RainaR2007}, the |
| 970 out-of-sample examples were used as a source of unsupervised data, and | |
| 971 experiments showed its positive effects in a \emph{limited labeled data} | |
| 972 scenario. However, many of the results by \citet{RainaR2007} (who used a | |
| 973 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught | |
| 974 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases. | |
| 975 We note instead that, for deep | |
| 976 architectures, our experiments show that such a positive effect is accomplished | |
| 977 even in a scenario with a \emph{large number of labeled examples}, | |
| 978 i.e., here, the relative gain of self-taught learning is probably preserved | |
| 979 in the asymptotic regime. | |
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980 |
| 582 | 981 {\bf Why would deep learners benefit more from the self-taught learning framework}? |
| 982 The key idea is that the lower layers of the predictor compute a hierarchy | |
| 983 of features that can be shared across tasks or across variants of the | |
|
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984 input distribution. A theoretical analysis of generalization improvements |
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985 due to sharing of intermediate features across tasks already points |
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986 towards that explanation~\cite{baxter95a}. |
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987 Intermediate features that can be used in different |
| 582 | 988 contexts can be estimated in a way that allows to share statistical |
| 989 strength. Features extracted through many levels are more likely to | |
| 990 be more abstract (as the experiments in~\citet{Goodfellow2009} suggest), | |
| 991 increasing the likelihood that they would be useful for a larger array | |
| 992 of tasks and input conditions. | |
| 993 Therefore, we hypothesize that both depth and unsupervised | |
| 994 pre-training play a part in explaining the advantages observed here, and future | |
| 995 experiments could attempt at teasing apart these factors. | |
| 996 And why would deep learners benefit from the self-taught learning | |
| 997 scenarios even when the number of labeled examples is very large? | |
| 998 We hypothesize that this is related to the hypotheses studied | |
| 999 in~\citet{Erhan+al-2010}. Whereas in~\citet{Erhan+al-2010} | |
| 1000 it was found that online learning on a huge dataset did not make the | |
| 1001 advantage of the deep learning bias vanish, a similar phenomenon | |
| 1002 may be happening here. We hypothesize that unsupervised pre-training | |
| 1003 of a deep hierarchy with self-taught learning initializes the | |
| 1004 model in the basin of attraction of supervised gradient descent | |
| 1005 that corresponds to better generalization. Furthermore, such good | |
| 1006 basins of attraction are not discovered by pure supervised learning | |
| 1007 (with or without self-taught settings), and more labeled examples | |
| 1008 does not allow the model to go from the poorer basins of attraction discovered | |
| 1009 by the purely supervised shallow models to the kind of better basins associated | |
| 1010 with deep learning and self-taught learning. | |
| 1011 | |
| 1012 A Flash demo of the recognizer (where both the MLP and the SDA can be compared) | |
| 1013 can be executed on-line at {\tt http://deep.host22.com}. | |
| 1014 | |
|
584
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1015 |
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1016 \section*{Appendix I: Detailed Numerical Results} |
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1017 |
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1018 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered. |
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1019 They also contain additional data such as test errors on P07 and standard errors. |
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1020 |
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1021 \begin{table}[ht] |
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1022 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + |
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1023 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training |
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1024 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture |
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1025 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) |
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1026 and using a validation set to select hyper-parameters and other training choices. |
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1027 \{SDA,MLP\}0 are trained on NIST, |
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1028 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. |
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1029 The human error rate on digits is a lower bound because it does not count digits that were |
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1030 recognized as letters. For comparison, the results found in the literature |
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1031 on NIST digits classification using the same test set are included.} |
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1032 \label{tab:sda-vs-mlp-vs-humans} |
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1033 \begin{center} |
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1034 \begin{tabular}{|l|r|r|r|r|} \hline |
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1035 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline |
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1036 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline |
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1037 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline |
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1038 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline |
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1039 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline |
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1040 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline |
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1041 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline |
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1042 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline |
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1043 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline |
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1044 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline |
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1045 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline |
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1046 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline |
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1047 \end{tabular} |
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1048 \end{center} |
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1049 \end{table} |
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1050 |
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1051 \begin{table}[ht] |
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1052 \caption{Relative change in error rates due to the use of perturbed training data, |
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1053 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. |
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1054 A positive value indicates that training on the perturbed data helped for the |
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1055 given test set (the first 3 columns on the 62-class tasks and the last one is |
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1056 on the clean 10-class digits). Clearly, the deep learning models did benefit more |
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1057 from perturbed training data, even when testing on clean data, whereas the MLP |
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1058 trained on perturbed data performed worse on the clean digits and about the same |
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1059 on the clean characters. } |
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1060 \label{tab:perturbation-effect} |
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1061 \begin{center} |
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1062 \begin{tabular}{|l|r|r|r|r|} \hline |
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1063 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline |
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1064 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline |
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1065 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline |
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1066 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline |
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1067 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline |
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1068 \end{tabular} |
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1069 \end{center} |
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1070 \end{table} |
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1071 |
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1072 \begin{table}[ht] |
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1073 \caption{Test error rates and relative change in error rates due to the use of |
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1074 a multi-task setting, i.e., training on each task in isolation vs training |
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1075 for all three tasks together, for MLPs vs SDAs. The SDA benefits much |
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1076 more from the multi-task setting. All experiments on only on the |
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1077 unperturbed NIST data, using validation error for model selection. |
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1078 Relative improvement is 1 - single-task error / multi-task error.} |
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1079 \label{tab:multi-task} |
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1080 \begin{center} |
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1081 \begin{tabular}{|l|r|r|r|} \hline |
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1082 & single-task & multi-task & relative \\ |
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1083 & setting & setting & improvement \\ \hline |
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1084 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline |
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1085 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline |
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1086 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline |
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1087 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline |
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1088 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline |
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1089 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline |
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1090 \end{tabular} |
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1091 \end{center} |
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1092 \end{table} |
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1093 |
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1094 %\afterpage{\clearpage} |
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1095 \clearpage |
| 582 | 1096 { |
|
583
ae77edb9df67
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1097 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter} |
| 582 | 1098 %\bibliographystyle{plainnat} |
| 1099 \bibliographystyle{unsrtnat} | |
| 1100 %\bibliographystyle{apalike} | |
| 1101 } | |
| 1102 | |
|
379
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1103 |
|
407
fe2e2964e7a3
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1104 \end{document} |