Mercurial > ift6266
annotate writeup/jmlr_submission.tex @ 625:128bc92897f2
Script to generate resized version of mnist in two different ways: rescaling and zeropadding
| author | Salah Rifai <salahmeister@gmail.com> |
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| date | Wed, 16 Mar 2011 00:05:39 -0400 |
| parents | 537f8b786655 |
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| rev | line source |
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1 %\documentclass[twoside,11pt]{article} % For LaTeX2e |
| 593 | 2 \documentclass{article} % For LaTeX2e |
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3 \usepackage{jmlr2e} |
| 593 | 4 \usepackage{times} |
| 5 \usepackage{wrapfig} | |
| 6 %\usepackage{amsthm} % not to be used with springer tools | |
| 7 \usepackage{amsmath} | |
| 8 \usepackage{bbm} | |
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9 \usepackage[utf8]{inputenc} |
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10 %\usepackage[psamsfonts]{amssymb} |
| 593 | 11 %\usepackage{algorithm,algorithmic} % not used after all |
| 12 \usepackage{graphicx,subfigure} | |
| 13 \usepackage{natbib} % was [numbers]{natbib} | |
| 14 | |
| 15 \addtolength{\textwidth}{10mm} | |
| 16 \addtolength{\evensidemargin}{-5mm} | |
| 17 \addtolength{\oddsidemargin}{-5mm} | |
| 18 | |
| 19 %\setlength\parindent{0mm} | |
| 20 | |
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21 \begin{document} |
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22 |
| 593 | 23 \title{Deep Self-Taught Learning for Handwritten Character Recognition} |
| 24 \author{ | |
| 25 Yoshua Bengio \and | |
| 26 Frédéric Bastien \and | |
| 27 Arnaud Bergeron \and | |
| 28 Nicolas Boulanger-Lewandowski \and | |
| 29 Thomas Breuel \and | |
| 30 Youssouf Chherawala \and | |
| 31 Moustapha Cisse \and | |
| 32 Myriam Côté \and | |
| 33 Dumitru Erhan \and | |
| 34 Jeremy Eustache \and | |
| 35 Xavier Glorot \and | |
| 36 Xavier Muller \and | |
| 37 Sylvain Pannetier Lebeuf \and | |
| 38 Razvan Pascanu \and | |
| 39 Salah Rifai \and | |
| 40 Francois Savard \and | |
| 41 Guillaume Sicard | |
| 42 } | |
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43 \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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44 \jmlrheading{}{2010}{}{10/2010}{XX/2011}{Yoshua Bengio et al} |
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45 \editor{} |
| 593 | 46 |
| 47 %\makeanontitle | |
| 48 \maketitle | |
| 49 | |
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50 {\bf Running title: Deep Self-Taught Learning} |
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51 |
| 593 | 52 %\vspace*{-2mm} |
| 53 \begin{abstract} | |
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54 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. 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 a large-scale handwritten character recognition setting. In fact, we show that they {\em beat previously published results and reach human-level performance}. |
| 593 | 55 \end{abstract} |
| 56 %\vspace*{-3mm} | |
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57 |
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58 \begin{keywords} |
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59 Deep learning, self-taught learning, out-of-distribution examples, handwritten character recognition, multi-task learning |
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60 \end{keywords} |
| 593 | 61 %\keywords{self-taught learning \and multi-task learning \and out-of-distribution examples \and deep learning \and handwriting recognition} |
| 62 | |
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63 |
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64 |
| 593 | 65 \section{Introduction} |
| 66 %\vspace*{-1mm} | |
| 67 | |
| 68 {\bf Deep Learning} has emerged as a promising new area of research in | |
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69 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. |
| 593 | 70 Learning algorithms for deep architectures are centered on the learning |
| 71 of useful representations of data, which are better suited to the task at hand, | |
| 72 and are organized in a hierarchy with multiple levels. | |
| 73 This is in part inspired by observations of the mammalian visual cortex, | |
| 74 which consists of a chain of processing elements, each of which is associated with a | |
| 75 different representation of the raw visual input. In fact, | |
| 76 it was found recently that the features learnt in deep architectures resemble | |
| 77 those observed in the first two of these stages (in areas V1 and V2 | |
| 78 of visual cortex) \citep{HonglakL2008}, and that they become more and | |
| 79 more invariant to factors of variation (such as camera movement) in | |
| 80 higher layers~\citep{Goodfellow2009}. | |
| 81 Learning a hierarchy of features increases the | |
| 82 ease and practicality of developing representations that are at once | |
| 83 tailored to specific tasks, yet are able to borrow statistical strength | |
| 84 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the | |
| 85 feature representation can lead to higher-level (more abstract, more | |
| 86 general) features that are more robust to unanticipated sources of | |
| 87 variance extant in real data. | |
| 88 | |
| 89 {\bf Self-taught learning}~\citep{RainaR2007} is a paradigm that combines principles | |
| 90 of semi-supervised and multi-task learning: the learner can exploit examples | |
| 91 that are unlabeled and possibly come from a distribution different from the target | |
| 92 distribution, e.g., from other classes than those of interest. | |
| 93 It has already been shown that deep learners can clearly take advantage of | |
| 94 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small}, | |
| 95 but more needs to be done to explore the impact | |
| 96 of {\em out-of-distribution} examples and of the {\em multi-task} setting | |
| 97 (one exception is~\citep{CollobertR2008}, which uses a different kind | |
| 98 of learning algorithm). In particular the {\em relative | |
| 99 advantage of deep learning} for these settings has not been evaluated. | |
| 100 The hypothesis discussed in the conclusion is that in the context of | |
| 101 multi-task learning and the availability of out-of-distribution training examples, | |
| 102 a deep hierarchy of features | |
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103 may be better able to provide {\em sharing of statistical strength} |
| 593 | 104 between different regions in input space or different tasks, compared to |
| 105 a shallow learner. | |
| 106 | |
| 107 Whereas a deep architecture can in principle be more powerful than a | |
| 108 shallow one in terms of representation, depth appears to render the | |
| 109 training problem more difficult in terms of optimization and local minima. | |
| 110 It is also only recently that successful algorithms were proposed to | |
| 111 overcome some of these difficulties. All are based on unsupervised | |
| 112 learning, often in an greedy layer-wise ``unsupervised pre-training'' | |
| 113 stage~\citep{Bengio-2009}. One of these layer initialization techniques, | |
| 114 applied here, is the Denoising | |
| 115 Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see Figure~\ref{fig:da}), | |
| 116 which | |
| 117 performed similarly or better than previously proposed Restricted Boltzmann | |
| 118 Machines in terms of unsupervised extraction of a hierarchy of features | |
| 119 useful for classification. Each layer is trained to denoise its | |
| 120 input, creating a layer of features that can be used as input for the next layer. | |
| 121 | |
| 122 %The principle is that each layer starting from | |
| 123 %the bottom is trained to encode its input (the output of the previous | |
| 124 %layer) and to reconstruct it from a corrupted version. After this | |
| 125 %unsupervised initialization, the stack of DAs can be | |
| 126 %converted into a deep supervised feedforward neural network and fine-tuned by | |
| 127 %stochastic gradient descent. | |
| 128 | |
| 129 % | |
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130 The {\bf main claim} of this paper is that deep learners (with several levels of representation) can |
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131 {\bf benefit more from self-taught learning than shallow learners} (with a single |
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132 level), both in the context of the multi-task setting and from {\em |
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133 out-of-distribution examples} in general. Because we are able to improve on state-of-the-art |
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134 performance and reach human-level performance |
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135 on a large-scale task, we consider that this paper is also a contribution |
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136 to advance the application of machine learning to handwritten character recognition. |
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137 More precisely, we ask and answer the following questions: |
| 593 | 138 |
| 139 %\begin{enumerate} | |
| 140 $\bullet$ %\item | |
| 141 Do the good results previously obtained with deep architectures on the | |
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142 MNIST digit images generalize to the setting of a similar but much larger and richer |
| 593 | 143 dataset, the NIST special database 19, with 62 classes and around 800k examples? |
| 144 | |
| 145 $\bullet$ %\item | |
| 146 To what extent does the perturbation of input images (e.g. adding | |
| 147 noise, affine transformations, background images) make the resulting | |
| 148 classifiers better not only on similarly perturbed images but also on | |
| 149 the {\em original clean examples}? We study this question in the | |
| 150 context of the 62-class and 10-class tasks of the NIST special database 19. | |
| 151 | |
| 152 $\bullet$ %\item | |
| 153 Do deep architectures {\em benefit {\bf more} from such out-of-distribution} | |
| 154 examples, i.e. do they benefit more from the self-taught learning~\citep{RainaR2007} framework? | |
| 155 We use highly perturbed examples to generate out-of-distribution examples. | |
| 156 | |
| 157 $\bullet$ %\item | |
| 158 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more} | |
| 159 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than | |
| 160 a corresponding shallow and purely supervised architecture? | |
| 161 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case) | |
| 162 to answer this question. | |
| 163 %\end{enumerate} | |
| 164 | |
| 165 Our experimental results provide positive evidence towards all of these questions, | |
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166 as well as {\em classifiers that reach human-level performance on 62-class isolated character |
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167 recognition and beat previously published results on the NIST dataset (special database 19)}. |
| 593 | 168 To achieve these results, we introduce in the next section a sophisticated system |
| 169 for stochastically transforming character images and then explain the methodology, | |
| 170 which is based on training with or without these transformed images and testing on | |
| 171 clean ones. We measure the relative advantage of out-of-distribution examples | |
| 172 (perturbed or out-of-class) | |
| 173 for a deep learner vs a supervised shallow one. | |
| 174 Code for generating these transformations as well as for the deep learning | |
| 175 algorithms are made available at {\tt http://hg.assembla.com/ift6266}. | |
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176 We also estimate the relative advantage for deep learners of training with |
| 593 | 177 other classes than those of interest, by comparing learners trained with |
| 178 62 classes with learners trained with only a subset (on which they | |
| 179 are then tested). | |
| 180 The conclusion discusses | |
| 181 the more general question of why deep learners may benefit so much from | |
| 182 the self-taught learning framework. Since out-of-distribution data | |
| 183 (perturbed or from other related classes) is very common, this conclusion | |
| 184 is of practical importance. | |
| 185 | |
| 186 %\vspace*{-3mm} | |
| 187 %\newpage | |
| 188 \section{Perturbed and Transformed Character Images} | |
| 189 \label{s:perturbations} | |
| 190 %\vspace*{-2mm} | |
| 191 | |
| 192 \begin{wrapfigure}[8]{l}{0.15\textwidth} | |
| 193 %\begin{minipage}[b]{0.14\linewidth} | |
| 194 %\vspace*{-5mm} | |
| 195 \begin{center} | |
| 196 \includegraphics[scale=.4]{images/Original.png}\\ | |
| 197 {\bf Original} | |
| 198 \end{center} | |
| 199 \end{wrapfigure} | |
| 200 %%\vspace{0.7cm} | |
| 201 %\end{minipage}% | |
| 202 %\hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 203 This section describes the different transformations we used to stochastically | |
| 204 transform $32 \times 32$ source images (such as the one on the left) | |
| 205 in order to obtain data from a larger distribution which | |
| 206 covers a domain substantially larger than the clean characters distribution from | |
| 207 which we start. | |
| 208 Although character transformations have been used before to | |
| 209 improve character recognizers, this effort is on a large scale both | |
| 210 in number of classes and in the complexity of the transformations, hence | |
| 211 in the complexity of the learning task. | |
| 212 The code for these transformations (mostly python) is available at | |
| 213 {\tt http://hg.assembla.com/ift6266}. All the modules in the pipeline share | |
| 214 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the | |
| 215 amount of deformation or noise introduced. | |
| 216 There are two main parts in the pipeline. The first one, | |
| 217 from slant to pinch below, performs transformations. The second | |
| 218 part, from blur to contrast, adds different kinds of noise. | |
| 219 %\end{minipage} | |
| 220 | |
| 221 %\vspace*{1mm} | |
| 222 \subsection{Transformations} | |
| 223 %{\large\bf 2.1 Transformations} | |
| 224 %\vspace*{1mm} | |
| 225 | |
| 226 \subsubsection*{Thickness} | |
| 227 | |
| 228 %\begin{wrapfigure}[7]{l}{0.15\textwidth} | |
| 229 \begin{minipage}[b]{0.14\linewidth} | |
| 230 %\centering | |
| 231 \begin{center} | |
| 232 \vspace*{-5mm} | |
| 233 \includegraphics[scale=.4]{images/Thick_only.png}\\ | |
| 234 %{\bf Thickness} | |
| 235 \end{center} | |
| 236 \vspace{.6cm} | |
| 237 \end{minipage}% | |
| 238 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 239 %\end{wrapfigure} | |
| 240 To change character {\bf thickness}, morphological operators of dilation and erosion~\citep{Haralick87,Serra82} | |
| 241 are applied. The neighborhood of each pixel is multiplied | |
| 242 element-wise with a {\em structuring element} matrix. | |
| 243 The pixel value is replaced by the maximum or the minimum of the resulting | |
| 244 matrix, respectively for dilation or erosion. Ten different structural elements with | |
| 245 increasing dimensions (largest is $5\times5$) were used. For each image, | |
| 246 randomly sample the operator type (dilation or erosion) with equal probability and one structural | |
| 247 element from a subset of the $n=round(m \times complexity)$ smallest structuring elements | |
| 248 where $m=10$ for dilation and $m=6$ for erosion (to avoid completely erasing thin characters). | |
| 249 A neutral element (no transformation) | |
| 250 is always present in the set. | |
| 251 %%\vspace{.4cm} | |
| 252 \end{minipage} | |
| 253 | |
| 254 \vspace{2mm} | |
| 255 | |
| 256 \subsubsection*{Slant} | |
| 257 \vspace*{2mm} | |
| 258 | |
| 259 \begin{minipage}[b]{0.14\linewidth} | |
| 260 \centering | |
| 261 \includegraphics[scale=.4]{images/Slant_only.png}\\ | |
| 262 %{\bf Slant} | |
| 263 \end{minipage}% | |
| 264 \hspace{0.3cm} | |
| 265 \begin{minipage}[b]{0.83\linewidth} | |
| 266 %\centering | |
| 267 To produce {\bf slant}, each row of the image is shifted | |
| 268 proportionally to its height: $shift = round(slant \times height)$. | |
| 269 $slant \sim U[-complexity,complexity]$. | |
| 270 The shift is randomly chosen to be either to the left or to the right. | |
| 271 \vspace{5mm} | |
| 272 \end{minipage} | |
| 273 %\vspace*{-4mm} | |
| 274 | |
| 275 %\newpage | |
| 276 | |
| 277 \subsubsection*{Affine Transformations} | |
| 278 | |
| 279 \begin{minipage}[b]{0.14\linewidth} | |
| 280 %\centering | |
| 281 %\begin{wrapfigure}[8]{l}{0.15\textwidth} | |
| 282 \begin{center} | |
| 283 \includegraphics[scale=.4]{images/Affine_only.png} | |
| 284 \vspace*{6mm} | |
| 285 %{\small {\bf Affine \mbox{Transformation}}} | |
| 286 \end{center} | |
| 287 %\end{wrapfigure} | |
| 288 \end{minipage}% | |
| 289 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 290 \noindent A $2 \times 3$ {\bf affine transform} matrix (with | |
| 291 parameters $(a,b,c,d,e,f)$) is sampled according to the $complexity$. | |
| 292 Output pixel $(x,y)$ takes the value of input pixel | |
| 293 nearest to $(ax+by+c,dx+ey+f)$, | |
| 294 producing scaling, translation, rotation and shearing. | |
| 295 Marginal distributions of $(a,b,c,d,e,f)$ have been tuned to | |
| 296 forbid large rotations (to avoid confusing classes) but to give good | |
| 297 variability of the transformation: $a$ and $d$ $\sim U[1-3 | |
| 298 complexity,1+3\,complexity]$, $b$ and $e$ $\sim U[-3 \,complexity,3\, | |
| 299 complexity]$, and $c$ and $f \sim U[-4 \,complexity, 4 \, | |
| 300 complexity]$.\\ | |
| 301 \end{minipage} | |
| 302 | |
| 303 %\vspace*{-4.5mm} | |
| 304 \subsubsection*{Local Elastic Deformations} | |
| 305 | |
| 306 %\begin{minipage}[t]{\linewidth} | |
| 307 %\begin{wrapfigure}[7]{l}{0.15\textwidth} | |
| 308 %\hspace*{-8mm} | |
| 309 \begin{minipage}[b]{0.14\linewidth} | |
| 310 %\centering | |
| 311 \begin{center} | |
| 312 \vspace*{5mm} | |
| 313 \includegraphics[scale=.4]{images/Localelasticdistorsions_only.png} | |
| 314 %{\bf Local Elastic Deformation} | |
| 315 \end{center} | |
| 316 %\end{wrapfigure} | |
| 317 \end{minipage}% | |
| 318 \hspace{3mm} | |
| 319 \begin{minipage}[b]{0.85\linewidth} | |
| 320 %%\vspace*{-20mm} | |
| 321 The {\bf local elastic deformation} | |
| 322 module induces a ``wiggly'' effect in the image, following~\citet{SimardSP03-short}, | |
| 323 which provides more details. | |
| 324 The intensity of the displacement fields is given by | |
| 325 $\alpha = \sqrt[3]{complexity} \times 10.0$, which are | |
| 326 convolved with a Gaussian 2D kernel (resulting in a blur) of | |
| 327 standard deviation $\sigma = 10 - 7 \times\sqrt[3]{complexity}$. | |
| 328 \vspace{2mm} | |
| 329 \end{minipage} | |
| 330 | |
| 331 \vspace*{4mm} | |
| 332 | |
| 333 \subsubsection*{Pinch} | |
| 334 | |
| 335 \begin{minipage}[b]{0.14\linewidth} | |
| 336 %\centering | |
| 337 %\begin{wrapfigure}[7]{l}{0.15\textwidth} | |
| 338 %\vspace*{-5mm} | |
| 339 \begin{center} | |
| 340 \includegraphics[scale=.4]{images/Pinch_only.png}\\ | |
| 341 \vspace*{15mm} | |
| 342 %{\bf Pinch} | |
| 343 \end{center} | |
| 344 %\end{wrapfigure} | |
| 345 %%\vspace{.6cm} | |
| 346 \end{minipage}% | |
| 347 \hspace{0.3cm}\begin{minipage}[b]{0.86\linewidth} | |
| 348 The {\bf pinch} module applies the ``Whirl and pinch'' GIMP filter with whirl set to 0. | |
| 349 A pinch is ``similar to projecting the image onto an elastic | |
| 350 surface and pressing or pulling on the center of the surface'' (GIMP documentation manual). | |
| 351 For a square input image, draw a radius-$r$ disk | |
| 352 around its center $C$. Any pixel $P$ belonging to | |
| 353 that disk has its value replaced by | |
| 354 the value of a ``source'' pixel in the original image, | |
| 355 on the line that goes through $C$ and $P$, but | |
| 356 at some other distance $d_2$. Define $d_1=distance(P,C)$ | |
| 357 and $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times | |
| 358 d_1$, where $pinch$ is a parameter of the filter. | |
| 359 The actual value is given by bilinear interpolation considering the pixels | |
| 360 around the (non-integer) source position thus found. | |
| 361 Here $pinch \sim U[-complexity, 0.7 \times complexity]$. | |
| 362 %%\vspace{1.5cm} | |
| 363 \end{minipage} | |
| 364 | |
| 365 %\vspace{1mm} | |
| 366 | |
| 367 %{\large\bf 2.2 Injecting Noise} | |
| 368 \subsection{Injecting Noise} | |
| 369 %\vspace{2mm} | |
| 370 | |
| 371 \subsubsection*{Motion Blur} | |
| 372 | |
| 373 %%\vspace*{-.2cm} | |
| 374 \begin{minipage}[t]{0.14\linewidth} | |
| 375 \centering | |
| 376 \vspace*{0mm} | |
| 377 \includegraphics[scale=.4]{images/Motionblur_only.png} | |
| 378 %{\bf Motion Blur} | |
| 379 \end{minipage}% | |
| 380 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 381 %%\vspace*{.5mm} | |
| 382 \vspace*{2mm} | |
| 383 The {\bf motion blur} module is GIMP's ``linear motion blur'', which | |
| 384 has parameters $length$ and $angle$. The value of | |
| 385 a pixel in the final image is approximately the mean of the first $length$ pixels | |
| 386 found by moving in the $angle$ direction, | |
| 387 $angle \sim U[0,360]$ degrees, and $length \sim {\rm Normal}(0,(3 \times complexity)^2)$. | |
| 388 %\vspace{5mm} | |
| 389 \end{minipage} | |
| 390 | |
| 391 %\vspace*{1mm} | |
| 392 | |
| 393 \subsubsection*{Occlusion} | |
| 394 | |
| 395 \begin{minipage}[t]{0.14\linewidth} | |
| 396 \centering | |
| 397 \vspace*{3mm} | |
| 398 \includegraphics[scale=.4]{images/occlusion_only.png}\\ | |
| 399 %{\bf Occlusion} | |
| 400 %%\vspace{.5cm} | |
| 401 \end{minipage}% | |
| 402 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 403 %\vspace*{-18mm} | |
| 404 The {\bf occlusion} module selects a random rectangle from an {\em occluder} character | |
| 405 image and places it over the original {\em occluded} | |
| 406 image. Pixels are combined by taking the max(occluder, occluded), | |
| 407 i.e. keeping the lighter ones. | |
| 408 The rectangle corners | |
| 409 are sampled so that larger complexity gives larger rectangles. | |
| 410 The destination position in the occluded image are also sampled | |
| 411 according to a normal distribution. | |
| 412 This module is skipped with probability 60\%. | |
| 413 %%\vspace{7mm} | |
| 414 \end{minipage} | |
| 415 | |
| 416 %\vspace*{1mm} | |
| 417 \subsubsection*{Gaussian Smoothing} | |
| 418 | |
| 419 %\begin{wrapfigure}[8]{l}{0.15\textwidth} | |
| 420 %\vspace*{-6mm} | |
| 421 \begin{minipage}[t]{0.14\linewidth} | |
| 422 \begin{center} | |
| 423 %\centering | |
| 424 \vspace*{6mm} | |
| 425 \includegraphics[scale=.4]{images/Bruitgauss_only.png} | |
| 426 %{\bf Gaussian Smoothing} | |
| 427 \end{center} | |
| 428 %\end{wrapfigure} | |
| 429 %%\vspace{.5cm} | |
| 430 \end{minipage}% | |
| 431 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth} | |
| 432 With the {\bf Gaussian smoothing} module, | |
| 433 different regions of the image are spatially smoothed. | |
| 434 This is achieved by first convolving | |
| 435 the image with an isotropic Gaussian kernel of | |
| 436 size and variance chosen uniformly in the ranges $[12,12 + 20 \times | |
| 437 complexity]$ and $[2,2 + 6 \times complexity]$. This filtered image is normalized | |
| 438 between $0$ and $1$. We also create an isotropic weighted averaging window, of the | |
| 439 kernel size, with maximum value at the center. For each image we sample | |
| 440 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be | |
| 441 averaging centers between the original image and the filtered one. We | |
| 442 initialize to zero a mask matrix of the image size. For each selected pixel | |
| 443 we add to the mask the averaging window centered on it. The final image is | |
| 444 computed from the following element-wise operation: $\frac{image + filtered\_image | |
| 445 \times mask}{mask+1}$. | |
| 446 This module is skipped with probability 75\%. | |
| 447 \end{minipage} | |
| 448 | |
| 449 %\newpage | |
| 450 | |
| 451 %\vspace*{-9mm} | |
| 452 \subsubsection*{Permute Pixels} | |
| 453 | |
| 454 %\hspace*{-3mm}\begin{minipage}[t]{0.18\linewidth} | |
| 455 %\centering | |
| 456 \begin{minipage}[t]{0.14\textwidth} | |
| 457 %\begin{wrapfigure}[7]{l}{ | |
| 458 %\vspace*{-5mm} | |
| 459 \begin{center} | |
| 460 \vspace*{1mm} | |
| 461 \includegraphics[scale=.4]{images/Permutpixel_only.png} | |
| 462 %{\small\bf Permute Pixels} | |
| 463 \end{center} | |
| 464 %\end{wrapfigure} | |
| 465 \end{minipage}% | |
| 466 \hspace{3mm}\begin{minipage}[t]{0.86\linewidth} | |
| 467 \vspace*{1mm} | |
| 468 %%\vspace*{-20mm} | |
| 469 This module {\bf permutes neighbouring pixels}. It first selects a | |
| 470 fraction $\frac{complexity}{3}$ of pixels randomly in the image. Each | |
| 471 of these pixels is then sequentially exchanged with a random pixel | |
| 472 among its four nearest neighbors (on its left, right, top or bottom). | |
| 473 This module is skipped with probability 80\%.\\ | |
| 474 %\vspace*{1mm} | |
| 475 \end{minipage} | |
| 476 | |
| 477 %\vspace{-3mm} | |
| 478 | |
| 479 \subsubsection*{Gaussian Noise} | |
| 480 | |
| 481 \begin{minipage}[t]{0.14\textwidth} | |
| 482 %\begin{wrapfigure}[7]{l}{ | |
| 483 %%\vspace*{-3mm} | |
| 484 \begin{center} | |
| 485 %\hspace*{-3mm}\begin{minipage}[t]{0.18\linewidth} | |
| 486 %\centering | |
| 487 \vspace*{0mm} | |
| 488 \includegraphics[scale=.4]{images/Distorsiongauss_only.png} | |
| 489 %{\small \bf Gauss. Noise} | |
| 490 \end{center} | |
| 491 %\end{wrapfigure} | |
| 492 \end{minipage}% | |
| 493 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth} | |
| 494 \vspace*{1mm} | |
| 495 %\vspace*{12mm} | |
| 496 The {\bf Gaussian noise} module simply adds, to each pixel of the image independently, a | |
| 497 noise $\sim Normal(0,(\frac{complexity}{10})^2)$. | |
| 498 This module is skipped with probability 70\%. | |
| 499 %%\vspace{1.1cm} | |
| 500 \end{minipage} | |
| 501 | |
| 502 %\vspace*{1.2cm} | |
| 503 | |
| 504 \subsubsection*{Background Image Addition} | |
| 505 | |
| 506 \begin{minipage}[t]{\linewidth} | |
| 507 \begin{minipage}[t]{0.14\linewidth} | |
| 508 \centering | |
| 509 \vspace*{0mm} | |
| 510 \includegraphics[scale=.4]{images/background_other_only.png} | |
| 511 %{\small \bf Bg Image} | |
| 512 \end{minipage}% | |
| 513 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 514 \vspace*{1mm} | |
| 515 Following~\citet{Larochelle-jmlr-2009}, the {\bf background image} module adds a random | |
| 516 background image behind the letter, from a randomly chosen natural image, | |
| 517 with contrast adjustments depending on $complexity$, to preserve | |
| 518 more or less of the original character image. | |
| 519 %%\vspace{.8cm} | |
| 520 \end{minipage} | |
| 521 \end{minipage} | |
| 522 %%\vspace{-.7cm} | |
| 523 | |
| 524 \subsubsection*{Salt and Pepper Noise} | |
| 525 | |
| 526 \begin{minipage}[t]{0.14\linewidth} | |
| 527 \centering | |
| 528 \vspace*{0mm} | |
| 529 \includegraphics[scale=.4]{images/Poivresel_only.png} | |
| 530 %{\small \bf Salt \& Pepper} | |
| 531 \end{minipage}% | |
| 532 \hspace{0.3cm}\begin{minipage}[t]{0.83\linewidth} | |
| 533 \vspace*{1mm} | |
| 534 The {\bf salt and pepper noise} module adds noise $\sim U[0,1]$ to random subsets of pixels. | |
| 535 The number of selected pixels is $0.2 \times complexity$. | |
| 536 This module is skipped with probability 75\%. | |
| 537 %%\vspace{.9cm} | |
| 538 \end{minipage} | |
| 539 %%\vspace{-.7cm} | |
| 540 | |
| 541 %\vspace{1mm} | |
| 542 \subsubsection*{Scratches} | |
| 543 | |
| 544 \begin{minipage}[t]{0.14\textwidth} | |
| 545 %\begin{wrapfigure}[7]{l}{ | |
| 546 %\begin{minipage}[t]{0.14\linewidth} | |
| 547 %\centering | |
| 548 \begin{center} | |
| 549 \vspace*{4mm} | |
| 550 %\hspace*{-1mm} | |
| 551 \includegraphics[scale=.4]{images/Rature_only.png}\\ | |
| 552 %{\bf Scratches} | |
| 553 \end{center} | |
| 554 \end{minipage}% | |
| 555 %\end{wrapfigure} | |
| 556 \hspace{0.3cm}\begin{minipage}[t]{0.86\linewidth} | |
| 557 %%\vspace{.4cm} | |
| 558 The {\bf scratches} module places line-like white patches on the image. The | |
| 559 lines are heavily transformed images of the digit ``1'' (one), chosen | |
| 560 at random among 500 such 1 images, | |
| 561 randomly cropped and rotated by an angle $\sim Normal(0,(100 \times | |
| 562 complexity)^2$ (in degrees), using bi-cubic interpolation. | |
| 563 Two passes of a grey-scale morphological erosion filter | |
| 564 are applied, reducing the width of the line | |
| 565 by an amount controlled by $complexity$. | |
| 566 This module is skipped with probability 85\%. The probabilities | |
| 567 of applying 1, 2, or 3 patches are (50\%,30\%,20\%). | |
| 568 \end{minipage} | |
| 569 | |
| 570 %\vspace*{1mm} | |
| 571 | |
| 572 \subsubsection*{Grey Level and Contrast Changes} | |
| 573 | |
| 574 \begin{minipage}[t]{0.15\linewidth} | |
| 575 \centering | |
| 576 \vspace*{0mm} | |
| 577 \includegraphics[scale=.4]{images/Contrast_only.png} | |
| 578 %{\bf Grey Level \& Contrast} | |
| 579 \end{minipage}% | |
| 580 \hspace{3mm}\begin{minipage}[t]{0.85\linewidth} | |
| 581 \vspace*{1mm} | |
| 582 The {\bf grey level and contrast} module changes the contrast by changing grey levels, and may invert the image polarity (white | |
| 583 to black and black to white). The contrast is $C \sim U[1-0.85 \times complexity,1]$ | |
| 584 so the image is normalized into $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The | |
| 585 polarity is inverted with probability 50\%. | |
| 586 %%\vspace{.7cm} | |
| 587 \end{minipage} | |
| 588 %\vspace{2mm} | |
| 589 | |
| 590 | |
| 591 \iffalse | |
| 592 \begin{figure}[ht] | |
| 593 \centerline{\resizebox{.9\textwidth}{!}{\includegraphics{images/example_t.png}}}\\ | |
| 594 \caption{Illustration of the pipeline of stochastic | |
| 595 transformations applied to the image of a lower-case \emph{t} | |
| 596 (the upper left image). Each image in the pipeline (going from | |
| 597 left to right, first top line, then bottom line) shows the result | |
| 598 of applying one of the modules in the pipeline. The last image | |
| 599 (bottom right) is used as training example.} | |
| 600 \label{fig:pipeline} | |
| 601 \end{figure} | |
| 602 \fi | |
| 603 | |
| 604 %\vspace*{-3mm} | |
| 605 \section{Experimental Setup} | |
| 606 %\vspace*{-1mm} | |
| 607 | |
| 608 Much previous work on deep learning had been performed on | |
| 609 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009}, | |
| 610 with 60~000 examples, and variants involving 10~000 | |
| 611 examples~\citep{Larochelle-jmlr-toappear-2008,VincentPLarochelleH2008}. | |
| 612 The focus here is on much larger training sets, from 10 times to | |
| 613 to 1000 times larger, and 62 classes. | |
| 614 | |
| 615 The first step in constructing the larger datasets (called NISTP and P07) is to sample from | |
| 616 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas}, | |
| 617 and {\bf OCR data} (scanned machine printed characters). Once a character | |
| 618 is sampled from one of these sources (chosen randomly), the second step is to | |
| 619 apply a pipeline of transformations and/or noise processes described in section \ref{s:perturbations}. | |
| 620 | |
| 621 To provide a baseline of error rate comparison we also estimate human performance | |
| 622 on both the 62-class task and the 10-class digits task. | |
| 623 We compare the best Multi-Layer Perceptrons (MLP) against | |
| 624 the best Stacked Denoising Auto-encoders (SDA), when | |
| 625 both models' hyper-parameters are selected to minimize the validation set error. | |
| 626 We also provide a comparison against a precise estimate | |
| 627 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 628 service (http://mturk.com). | |
| 629 AMT users are paid small amounts | |
| 630 of money to perform tasks for which human intelligence is required. | |
| 631 Mechanical Turk has been used extensively in natural language processing and vision. | |
| 632 %processing \citep{SnowEtAl2008} and vision | |
| 633 %\citep{SorokinAndForsyth2008,whitehill09}. | |
| 634 AMT users were presented | |
| 635 with 10 character images (from a test set) and asked to choose 10 corresponding ASCII | |
| 636 characters. They were forced to choose a single character class (either among the | |
| 637 62 or 10 character classes) for each image. | |
| 638 80 subjects classified 2500 images per (dataset,task) pair. | |
| 639 Different humans labelers sometimes provided a different label for the same | |
| 640 example, and we were able to estimate the error variance due to this effect | |
| 641 because each image was classified by 3 different persons. | |
| 642 The average error of humans on the 62-class task NIST test set | |
| 643 is 18.2\%, with a standard error of 0.1\%. | |
| 644 | |
| 645 %\vspace*{-3mm} | |
| 646 \subsection{Data Sources} | |
| 647 %\vspace*{-2mm} | |
| 648 | |
| 649 %\begin{itemize} | |
| 650 %\item | |
| 651 {\bf NIST.} | |
| 652 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995}, | |
| 653 widely used for training and testing character | |
| 654 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}. | |
| 655 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications, | |
| 656 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes | |
| 657 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity. | |
| 658 The fourth partition (called $hsf_4$, 82587 examples), | |
| 659 experimentally recognized to be the most difficult one, is the one recommended | |
| 660 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} | |
| 661 for that purpose. We randomly split the remainder (731668 examples) into a training set and a validation set for | |
| 662 model selection. | |
| 663 The performances reported by previous work on that dataset mostly use only the digits. | |
| 664 Here we use all the classes both in the training and testing phase. This is especially | |
| 665 useful to estimate the effect of a multi-task setting. | |
| 666 The distribution of the classes in the NIST training and test sets differs | |
| 667 substantially, with relatively many more digits in the test set, and a more uniform distribution | |
| 668 of letters in the test set (whereas in the training set they are distributed | |
| 669 more like in natural text). | |
| 670 %\vspace*{-1mm} | |
| 671 | |
| 672 %\item | |
| 673 {\bf Fonts.} | |
| 674 In order to have a good variety of sources we downloaded an important number of free fonts from: | |
| 675 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}. | |
| 676 % TODO: pointless to anonymize, it's not pointing to our work | |
| 677 Including the operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from. | |
| 678 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, | |
| 679 directly as input to our models. | |
| 680 %\vspace*{-1mm} | |
| 681 | |
| 682 %\item | |
| 683 {\bf Captchas.} | |
| 684 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for | |
| 685 generating characters of the same format as the NIST dataset. This software is based on | |
| 686 a random character class generator and various kinds of transformations similar to those described in the previous sections. | |
| 687 In order to increase the variability of the data generated, many different fonts are used for generating the characters. | |
| 688 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity | |
| 689 depending on the value of the complexity parameter provided by the user of the data source. | |
| 690 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class? | |
| 691 %\vspace*{-1mm} | |
| 692 | |
| 693 %\item | |
| 694 {\bf OCR data.} | |
| 695 A large set (2 million) of scanned, OCRed and manually verified machine-printed | |
| 696 characters where included as an | |
| 697 additional source. This set is part of a larger corpus being collected by the Image Understanding | |
| 698 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern | |
| 699 ({\tt http://www.iupr.com}), and which will be publicly released. | |
| 700 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this | |
| 701 %\end{itemize} | |
| 702 | |
| 703 %\vspace*{-3mm} | |
| 704 \subsection{Data Sets} | |
| 705 %\vspace*{-2mm} | |
| 706 | |
| 707 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label | |
| 708 from one of the 62 character classes. | |
| 709 %\begin{itemize} | |
| 710 %\vspace*{-1mm} | |
| 711 | |
| 712 %\item | |
| 713 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has | |
| 714 \{651668 / 80000 / 82587\} \{training / validation / test\} examples. | |
| 715 %\vspace*{-1mm} | |
| 716 | |
| 717 %\item | |
| 718 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources | |
| 719 and sending them through the transformation pipeline described in section \ref{s:perturbations}. | |
| 720 For each new example to generate, a data source is selected with probability $10\%$ from the fonts, | |
| 721 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the | |
| 722 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$. | |
| 723 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples. | |
| 724 %\vspace*{-1mm} | |
| 725 | |
| 726 %\item | |
| 727 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources) | |
| 728 except that we only apply | |
| 729 transformations from slant to pinch. Therefore, the character is | |
| 730 transformed but no additional noise is added to the image, giving images | |
| 731 closer to the NIST dataset. | |
| 732 It has \{81920000 / 80000 / 20000\} \{training / validation / test\} examples. | |
| 733 %\end{itemize} | |
| 734 | |
| 735 %\vspace*{-3mm} | |
| 736 \subsection{Models and their Hyperparameters} | |
| 737 %\vspace*{-2mm} | |
| 738 | |
| 739 The experiments are performed using MLPs (with a single | |
| 740 hidden layer) and SDAs. | |
| 741 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.} | |
| 742 | |
| 743 {\bf Multi-Layer Perceptrons (MLP).} | |
| 744 Whereas previous work had compared deep architectures to both shallow MLPs and | |
| 745 SVMs, we only compared to MLPs here because of the very large datasets used | |
| 746 (making the use of SVMs computationally challenging because of their quadratic | |
| 747 scaling behavior). Preliminary experiments on training SVMs (libSVM) with subsets of the training | |
| 748 set allowing the program to fit in memory yielded substantially worse results | |
| 749 than those obtained with MLPs. For training on nearly a billion examples | |
| 750 (with the perturbed data), the MLPs and SDA are much more convenient than | |
| 751 classifiers based on kernel methods. | |
| 752 The MLP has a single hidden layer with $\tanh$ activation functions, and softmax (normalized | |
| 753 exponentials) on the output layer for estimating $P(class | image)$. | |
| 754 The number of hidden units is taken in $\{300,500,800,1000,1500\}$. | |
| 755 Training examples are presented in minibatches of size 20. A constant learning | |
| 756 rate was chosen among $\{0.001, 0.01, 0.025, 0.075, 0.1, 0.5\}$. | |
| 757 %through preliminary experiments (measuring performance on a validation set), | |
| 758 %and $0.1$ (which was found to work best) was then selected for optimizing on | |
| 759 %the whole training sets. | |
| 760 %\vspace*{-1mm} | |
| 761 | |
| 762 | |
| 763 {\bf Stacked Denoising Auto-Encoders (SDA).} | |
| 764 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs) | |
| 765 can be used to initialize the weights of each layer of a deep MLP (with many hidden | |
| 766 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006}, | |
| 767 apparently setting parameters in the | |
| 768 basin of attraction of supervised gradient descent yielding better | |
| 769 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised | |
| 770 pre-training phase} uses all of the training images but not the training labels. | |
| 771 Each layer is trained in turn to produce a new representation of its input | |
| 772 (starting from the raw pixels). | |
| 773 It is hypothesized that the | |
| 774 advantage brought by this procedure stems from a better prior, | |
| 775 on the one hand taking advantage of the link between the input | |
| 776 distribution $P(x)$ and the conditional distribution of interest | |
| 777 $P(y|x)$ (like in semi-supervised learning), and on the other hand | |
| 778 taking advantage of the expressive power and bias implicit in the | |
| 779 deep architecture (whereby complex concepts are expressed as | |
| 780 compositions of simpler ones through a deep hierarchy). | |
| 781 | |
| 782 \begin{figure}[ht] | |
| 783 %\vspace*{-2mm} | |
| 784 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}} | |
| 785 %\vspace*{-2mm} | |
| 786 \caption{Illustration of the computations and training criterion for the denoising | |
| 787 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of | |
| 788 the layer (i.e. raw input or output of previous layer) | |
| 789 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$. | |
| 790 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which | |
| 791 is compared to the uncorrupted input $x$ through the loss function | |
| 792 $L_H(x,z)$, whose expected value is approximately minimized during training | |
| 793 by tuning $\theta$ and $\theta'$.} | |
| 794 \label{fig:da} | |
| 795 %\vspace*{-2mm} | |
| 796 \end{figure} | |
| 797 | |
| 798 Here we chose to use the Denoising | |
| 799 Auto-encoder~\citep{VincentPLarochelleH2008} as the building block for | |
| 800 these deep hierarchies of features, as it is simple to train and | |
| 801 explain (see Figure~\ref{fig:da}, as well as | |
| 802 tutorial and code there: {\tt http://deeplearning.net/tutorial}), | |
| 803 provides efficient inference, and yielded results | |
| 804 comparable or better than RBMs in series of experiments | |
| 805 \citep{VincentPLarochelleH2008}. During training, a Denoising | |
| 806 Auto-encoder is presented with a stochastically corrupted version | |
| 807 of the input and trained to reconstruct the uncorrupted input, | |
| 808 forcing the hidden units to represent the leading regularities in | |
| 809 the data. Here we use the random binary masking corruption | |
| 810 (which sets to 0 a random subset of the inputs). | |
| 811 Once it is trained, in a purely unsupervised way, | |
| 812 its hidden units' activations can | |
| 813 be used as inputs for training a second one, etc. | |
| 814 After this unsupervised pre-training stage, the parameters | |
| 815 are used to initialize a deep MLP, which is fine-tuned by | |
| 816 the same standard procedure used to train them (see previous section). | |
| 817 The SDA hyper-parameters are the same as for the MLP, with the addition of the | |
| 818 amount of corruption noise (we used the masking noise process, whereby a | |
| 819 fixed proportion of the input values, randomly selected, are zeroed), and a | |
| 820 separate learning rate for the unsupervised pre-training stage (selected | |
| 821 from the same above set). The fraction of inputs corrupted was selected | |
| 822 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number | |
| 823 of hidden layers but it was fixed to 3 based on previous work with | |
| 824 SDAs on MNIST~\citep{VincentPLarochelleH2008}. The size of the hidden | |
| 825 layers was kept constant across hidden layers, and the best results | |
| 826 were obtained with the largest values that we could experiment | |
| 827 with given our patience, with 1000 hidden units. | |
| 828 | |
| 829 %\vspace*{-1mm} | |
| 830 | |
| 831 \begin{figure}[ht] | |
| 832 %\vspace*{-2mm} | |
| 833 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}} | |
| 834 %\vspace*{-3mm} | |
| 835 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained | |
| 836 on NIST, 1 on NISTP, and 2 on P07. Left: overall results | |
| 837 of all models, on NIST and NISTP test sets. | |
| 838 Right: error rates on NIST test digits only, along with the previous results from | |
| 839 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005} | |
| 840 respectively based on ART, nearest neighbors, MLPs, and SVMs.} | |
| 841 \label{fig:error-rates-charts} | |
| 842 %\vspace*{-2mm} | |
| 843 \end{figure} | |
| 844 | |
| 845 | |
| 846 \begin{figure}[ht] | |
| 847 %\vspace*{-3mm} | |
| 848 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}} | |
| 849 %\vspace*{-3mm} | |
| 850 \caption{Relative improvement in error rate due to self-taught learning. | |
| 851 Left: Improvement (or loss, when negative) | |
| 852 induced by out-of-distribution examples (perturbed data). | |
| 853 Right: Improvement (or loss, when negative) induced by multi-task | |
| 854 learning (training on all classes and testing only on either digits, | |
| 855 upper case, or lower-case). The deep learner (SDA) benefits more from | |
| 856 both self-taught learning scenarios, compared to the shallow MLP.} | |
| 857 \label{fig:improvements-charts} | |
| 858 %\vspace*{-2mm} | |
| 859 \end{figure} | |
| 860 | |
| 861 \section{Experimental Results} | |
| 862 %\vspace*{-2mm} | |
| 863 | |
| 864 %%\vspace*{-1mm} | |
| 865 %\subsection{SDA vs MLP vs Humans} | |
| 866 %%\vspace*{-1mm} | |
| 867 The models are either trained on NIST (MLP0 and SDA0), | |
| 868 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested | |
| 869 on either NIST, NISTP or P07, either on the 62-class task | |
| 870 or on the 10-digits task. Training (including about half | |
| 871 for unsupervised pre-training, for DAs) on the larger | |
| 872 datasets takes around one day on a GPU-285. | |
| 873 Figure~\ref{fig:error-rates-charts} summarizes the results obtained, | |
| 874 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1, | |
| 875 SDA2), along with the previous results on the digits NIST special database | |
| 876 19 test set from the literature, respectively based on ARTMAP neural | |
| 877 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search | |
| 878 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs | |
| 879 ~\citep{Milgram+al-2005}. More detailed and complete numerical results | |
| 880 (figures and tables, including standard errors on the error rates) can be | |
| 881 found in Appendix. | |
| 882 The deep learner not only outperformed the shallow ones and | |
| 883 previously published performance (in a statistically and qualitatively | |
| 884 significant way) but when trained with perturbed data | |
| 885 reaches human performance on both the 62-class task | |
| 886 and the 10-class (digits) task. | |
| 887 17\% error (SDA1) or 18\% error (humans) may seem large but a large | |
| 888 majority of the errors from humans and from SDA1 are from out-of-context | |
| 889 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a | |
| 890 ``c'' and a ``C'' are often indistinguishible). | |
| 891 | |
| 892 In addition, as shown in the left of | |
| 893 Figure~\ref{fig:improvements-charts}, the relative improvement in error | |
| 894 rate brought by self-taught learning is greater for the SDA, and these | |
| 895 differences with the MLP are statistically and qualitatively | |
| 896 significant. | |
| 897 The left side of the figure shows the improvement to the clean | |
| 898 NIST test set error brought by the use of out-of-distribution examples | |
| 899 (i.e. the perturbed examples examples from NISTP or P07). | |
| 900 Relative percent change is measured by taking | |
| 901 $100 \% \times$ (original model's error / perturbed-data model's error - 1). | |
| 902 The right side of | |
| 903 Figure~\ref{fig:improvements-charts} shows the relative improvement | |
| 904 brought by the use of a multi-task setting, in which the same model is | |
| 905 trained for more classes than the target classes of interest (i.e. training | |
| 906 with all 62 classes when the target classes are respectively the digits, | |
| 907 lower-case, or upper-case characters). Again, whereas the gain from the | |
| 908 multi-task setting is marginal or negative for the MLP, it is substantial | |
| 909 for the SDA. Note that to simplify these multi-task experiments, only the original | |
| 910 NIST dataset is used. For example, the MLP-digits bar shows the relative | |
| 911 percent improvement in MLP error rate on the NIST digits test set | |
| 912 is $100\% \times$ (single-task | |
| 913 model's error / multi-task model's error - 1). The single-task model is | |
| 914 trained with only 10 outputs (one per digit), seeing only digit examples, | |
| 915 whereas the multi-task model is trained with 62 outputs, with all 62 | |
| 916 character classes as examples. Hence the hidden units are shared across | |
| 917 all tasks. For the multi-task model, the digit error rate is measured by | |
| 918 comparing the correct digit class with the output class associated with the | |
| 919 maximum conditional probability among only the digit classes outputs. The | |
| 920 setting is similar for the other two target classes (lower case characters | |
| 921 and upper case characters). | |
| 922 %%\vspace*{-1mm} | |
| 923 %\subsection{Perturbed Training Data More Helpful for SDA} | |
| 924 %%\vspace*{-1mm} | |
| 925 | |
| 926 %%\vspace*{-1mm} | |
| 927 %\subsection{Multi-Task Learning Effects} | |
| 928 %%\vspace*{-1mm} | |
| 929 | |
| 930 \iffalse | |
| 931 As previously seen, the SDA is better able to benefit from the | |
| 932 transformations applied to the data than the MLP. In this experiment we | |
| 933 define three tasks: recognizing digits (knowing that the input is a digit), | |
| 934 recognizing upper case characters (knowing that the input is one), and | |
| 935 recognizing lower case characters (knowing that the input is one). We | |
| 936 consider the digit classification task as the target task and we want to | |
| 937 evaluate whether training with the other tasks can help or hurt, and | |
| 938 whether the effect is different for MLPs versus SDAs. The goal is to find | |
| 939 out if deep learning can benefit more (or less) from multiple related tasks | |
| 940 (i.e. the multi-task setting) compared to a corresponding purely supervised | |
| 941 shallow learner. | |
| 942 | |
| 943 We use a single hidden layer MLP with 1000 hidden units, and a SDA | |
| 944 with 3 hidden layers (1000 hidden units per layer), pre-trained and | |
| 945 fine-tuned on NIST. | |
| 946 | |
| 947 Our results show that the MLP benefits marginally from the multi-task setting | |
| 948 in the case of digits (5\% relative improvement) but is actually hurt in the case | |
| 949 of characters (respectively 3\% and 4\% worse for lower and upper class characters). | |
| 950 On the other hand the SDA benefited from the multi-task setting, with relative | |
| 951 error rate improvements of 27\%, 15\% and 13\% respectively for digits, | |
| 952 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. | |
| 953 \fi | |
| 954 | |
| 955 | |
| 956 %\vspace*{-2mm} | |
| 957 \section{Conclusions and Discussion} | |
| 958 %\vspace*{-2mm} | |
| 959 | |
| 960 We have found that the self-taught learning framework is more beneficial | |
| 961 to a deep learner than to a traditional shallow and purely | |
| 962 supervised learner. More precisely, | |
| 963 the answers are positive for all the questions asked in the introduction. | |
| 964 %\begin{itemize} | |
| 965 | |
| 966 $\bullet$ %\item | |
| 967 {\bf Do the good results previously obtained with deep architectures on the | |
| 968 MNIST digits generalize to a much larger and richer (but similar) | |
| 969 dataset, the NIST special database 19, with 62 classes and around 800k examples}? | |
| 970 Yes, the SDA {\em systematically outperformed the MLP and all the previously | |
| 971 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level | |
| 972 performance} at around 17\% error on the 62-class task and 1.4\% on the digits, | |
| 973 and beating previously published results on the same data. | |
| 974 | |
| 975 $\bullet$ %\item | |
| 976 {\bf To what extent do self-taught learning scenarios help deep learners, | |
| 977 and do they help them more than shallow supervised ones}? | |
| 978 We found that distorted training examples not only made the resulting | |
| 979 classifier better on similarly perturbed images but also on | |
| 980 the {\em original clean examples}, and more importantly and more novel, | |
| 981 that deep architectures benefit more from such {\em out-of-distribution} | |
| 982 examples. MLPs were helped by perturbed training examples when tested on perturbed input | |
| 983 images (65\% relative improvement on NISTP) | |
| 984 but only marginally helped (5\% relative improvement on all classes) | |
| 985 or even hurt (10\% relative loss on digits) | |
| 986 with respect to clean examples . On the other hand, the deep SDAs | |
| 987 were significantly boosted by these out-of-distribution examples. | |
| 988 Similarly, whereas the improvement due to the multi-task setting was marginal or | |
| 989 negative for the MLP (from +5.6\% to -3.6\% relative change), | |
| 990 it was quite significant for the SDA (from +13\% to +27\% relative change), | |
| 991 which may be explained by the arguments below. | |
| 992 %\end{itemize} | |
| 993 | |
| 994 In the original self-taught learning framework~\citep{RainaR2007}, the | |
| 995 out-of-sample examples were used as a source of unsupervised data, and | |
| 996 experiments showed its positive effects in a \emph{limited labeled data} | |
| 997 scenario. However, many of the results by \citet{RainaR2007} (who used a | |
| 998 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught | |
| 999 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases. | |
| 1000 We note instead that, for deep | |
| 1001 architectures, our experiments show that such a positive effect is accomplished | |
| 1002 even in a scenario with a \emph{large number of labeled examples}, | |
| 1003 i.e., here, the relative gain of self-taught learning is probably preserved | |
| 1004 in the asymptotic regime. | |
| 1005 | |
| 1006 {\bf Why would deep learners benefit more from the self-taught learning framework}? | |
| 1007 The key idea is that the lower layers of the predictor compute a hierarchy | |
| 1008 of features that can be shared across tasks or across variants of the | |
| 1009 input distribution. A theoretical analysis of generalization improvements | |
| 1010 due to sharing of intermediate features across tasks already points | |
| 1011 towards that explanation~\cite{baxter95a}. | |
| 1012 Intermediate features that can be used in different | |
| 1013 contexts can be estimated in a way that allows to share statistical | |
| 1014 strength. Features extracted through many levels are more likely to | |
|
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1015 be more abstract and more invariant to some of the factors of variation |
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1016 in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest), |
| 593 | 1017 increasing the likelihood that they would be useful for a larger array |
| 1018 of tasks and input conditions. | |
| 1019 Therefore, we hypothesize that both depth and unsupervised | |
| 1020 pre-training play a part in explaining the advantages observed here, and future | |
| 1021 experiments could attempt at teasing apart these factors. | |
| 1022 And why would deep learners benefit from the self-taught learning | |
| 1023 scenarios even when the number of labeled examples is very large? | |
| 1024 We hypothesize that this is related to the hypotheses studied | |
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1025 in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010} |
| 593 | 1026 it was found that online learning on a huge dataset did not make the |
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1027 advantage of the deep learning bias vanish, and a similar phenomenon |
| 593 | 1028 may be happening here. We hypothesize that unsupervised pre-training |
| 1029 of a deep hierarchy with self-taught learning initializes the | |
| 1030 model in the basin of attraction of supervised gradient descent | |
| 1031 that corresponds to better generalization. Furthermore, such good | |
| 1032 basins of attraction are not discovered by pure supervised learning | |
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1033 (with or without self-taught settings) from random initialization, and more labeled examples |
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1034 does not allow the shallow or purely supervised models to discover |
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1035 the kind of better basins associated |
| 593 | 1036 with deep learning and self-taught learning. |
| 1037 | |
| 1038 A Flash demo of the recognizer (where both the MLP and the SDA can be compared) | |
| 1039 can be executed on-line at {\tt http://deep.host22.com}. | |
| 1040 | |
| 1041 | |
| 1042 \section*{Appendix I: Detailed Numerical Results} | |
| 1043 | |
| 1044 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered. | |
| 1045 They also contain additional data such as test errors on P07 and standard errors. | |
| 1046 | |
| 1047 \begin{table}[ht] | |
| 1048 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + | |
| 1049 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training | |
| 1050 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture | |
| 1051 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) | |
| 1052 and using a validation set to select hyper-parameters and other training choices. | |
| 1053 \{SDA,MLP\}0 are trained on NIST, | |
| 1054 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. | |
| 1055 The human error rate on digits is a lower bound because it does not count digits that were | |
| 1056 recognized as letters. For comparison, the results found in the literature | |
| 1057 on NIST digits classification using the same test set are included.} | |
| 1058 \label{tab:sda-vs-mlp-vs-humans} | |
| 1059 \begin{center} | |
| 1060 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 1061 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 1062 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline | |
| 1063 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline | |
| 1064 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline | |
| 1065 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline | |
| 1066 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline | |
| 1067 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline | |
| 1068 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline | |
| 1069 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline | |
| 1070 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline | |
| 1071 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline | |
| 1072 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline | |
| 1073 \end{tabular} | |
| 1074 \end{center} | |
| 1075 \end{table} | |
| 1076 | |
| 1077 \begin{table}[ht] | |
| 1078 \caption{Relative change in error rates due to the use of perturbed training data, | |
| 1079 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. | |
| 1080 A positive value indicates that training on the perturbed data helped for the | |
| 1081 given test set (the first 3 columns on the 62-class tasks and the last one is | |
| 1082 on the clean 10-class digits). Clearly, the deep learning models did benefit more | |
| 1083 from perturbed training data, even when testing on clean data, whereas the MLP | |
| 1084 trained on perturbed data performed worse on the clean digits and about the same | |
| 1085 on the clean characters. } | |
| 1086 \label{tab:perturbation-effect} | |
| 1087 \begin{center} | |
| 1088 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 1089 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 1090 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline | |
| 1091 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline | |
| 1092 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline | |
| 1093 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline | |
| 1094 \end{tabular} | |
| 1095 \end{center} | |
| 1096 \end{table} | |
| 1097 | |
| 1098 \begin{table}[ht] | |
| 1099 \caption{Test error rates and relative change in error rates due to the use of | |
| 1100 a multi-task setting, i.e., training on each task in isolation vs training | |
| 1101 for all three tasks together, for MLPs vs SDAs. The SDA benefits much | |
| 1102 more from the multi-task setting. All experiments on only on the | |
| 1103 unperturbed NIST data, using validation error for model selection. | |
| 1104 Relative improvement is 1 - single-task error / multi-task error.} | |
| 1105 \label{tab:multi-task} | |
| 1106 \begin{center} | |
| 1107 \begin{tabular}{|l|r|r|r|} \hline | |
| 1108 & single-task & multi-task & relative \\ | |
| 1109 & setting & setting & improvement \\ \hline | |
| 1110 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline | |
| 1111 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline | |
| 1112 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline | |
| 1113 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline | |
| 1114 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline | |
| 1115 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline | |
| 1116 \end{tabular} | |
| 1117 \end{center} | |
| 1118 \end{table} | |
| 1119 | |
| 1120 %\afterpage{\clearpage} | |
| 1121 \clearpage | |
| 1122 { | |
| 1123 %\bibliographystyle{spbasic} % basic style, author-year citations | |
| 1124 \bibliographystyle{plainnat} | |
| 1125 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter} | |
| 1126 %\bibliographystyle{unsrtnat} | |
| 1127 %\bibliographystyle{apalike} | |
| 1128 } | |
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| 1131 \end{document} |