Mercurial > ift6266
annotate writeup/aistats2011_cameraready.tex @ 639:507cb92d8e15
modifs mineures
| author | Yoshua Bengio <bengioy@iro.umontreal.ca> |
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| date | Sun, 20 Mar 2011 16:49:44 -0400 |
| parents | 677d1b1d8158 |
| children | 8b1a0b9fecff |
| rev | line source |
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| 627 | 1 %\documentclass[twoside,11pt]{article} % For LaTeX2e |
| 2 \documentclass{article} % For LaTeX2e | |
| 3 \usepackage[accepted]{aistats2e_2011} | |
| 4 %\usepackage{times} | |
| 5 \usepackage{wrapfig} | |
| 6 \usepackage{amsthm} | |
| 7 \usepackage{amsmath} | |
| 8 \usepackage{bbm} | |
| 9 \usepackage[utf8]{inputenc} | |
| 10 \usepackage[psamsfonts]{amssymb} | |
| 11 %\usepackage{algorithm,algorithmic} % not used after all | |
| 12 \usepackage{graphicx,subfigure} | |
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13 \usepackage{natbib} |
| 639 | 14 %\usepackage{afterpage} |
| 627 | 15 |
| 16 \addtolength{\textwidth}{10mm} | |
| 17 \addtolength{\evensidemargin}{-5mm} | |
| 18 \addtolength{\oddsidemargin}{-5mm} | |
| 19 | |
| 20 %\setlength\parindent{0mm} | |
| 21 | |
| 22 \begin{document} | |
| 23 | |
| 24 \twocolumn[ | |
| 25 \aistatstitle{Deep Learners Benefit More from Out-of-Distribution Examples} | |
| 26 \runningtitle{Deep Learners for Out-of-Distribution Examples} | |
| 27 \runningauthor{Bengio et. al.} | |
| 28 \aistatsauthor{ | |
| 29 Yoshua Bengio \and | |
| 30 Frédéric Bastien \and | |
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31 \bf Arnaud Bergeron \and |
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32 Nicolas Boulanger-Lewandowski \and \\ |
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33 \bf Thomas Breuel \and |
| 627 | 34 Youssouf Chherawala \and |
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35 \bf Moustapha Cisse \and |
| 639 | 36 Myriam Côté \and |
| 37 \bf Dumitru Erhan \\ | |
| 38 \and \bf Jeremy Eustache \and | |
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39 \bf Xavier Glorot \and |
| 639 | 40 Xavier Muller \and |
| 41 \bf Sylvain Pannetier Lebeuf \\ | |
| 42 \and \bf Razvan Pascanu \and | |
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43 \bf Salah Rifai \and |
| 639 | 44 Francois Savard \and |
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45 \bf Guillaume Sicard \\ |
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46 \vspace*{1mm}} |
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47 |
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48 %I can't use aistatsaddress in a single side paragraphe. |
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49 %The document is 2 colums, but this section span the 2 colums, sot there is only 1 left |
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50 \center{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} |
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51 \vspace*{5mm} |
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52 ] |
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53 %\aistatsaddress{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} |
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54 |
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55 |
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56 %\vspace*{5mm}} |
| 627 | 57 %\date{{\tt bengioy@iro.umontreal.ca}, Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} |
| 58 %\jmlrheading{}{2010}{}{10/2010}{XX/2011}{Yoshua Bengio et al} | |
| 59 %\editor{} | |
| 60 | |
| 61 %\makeanontitle | |
| 62 %\maketitle | |
| 63 | |
| 64 %{\bf Running title: Deep Self-Taught Learning} | |
| 65 | |
| 66 \vspace*{5mm} | |
| 67 \begin{abstract} | |
| 68 Recent theoretical and empirical work in statistical machine learning has demonstrated the potential of learning algorithms for deep architectures, i.e., function classes obtained by composing multiple levels of representation. The hypothesis evaluated here is that intermediate levels of representation, because they can be shared across tasks and examples from different but related distributions, can yield even more benefits. Comparative experiments were performed on a large-scale handwritten character recognition setting with 62 classes (upper case, lower case, digits), using both a multi-task setting and perturbed examples in order to obtain out-of-distribution examples. The results agree with the hypothesis, and show that a deep learner did {\em beat previously published results and reached human-level performance}. | |
| 69 \end{abstract} | |
| 70 %\vspace*{-3mm} | |
| 71 | |
| 72 %\begin{keywords} | |
| 73 %Deep learning, self-taught learning, out-of-distribution examples, handwritten character recognition, multi-task learning | |
| 74 %\end{keywords} | |
| 75 %\keywords{self-taught learning \and multi-task learning \and out-of-distribution examples \and deep learning \and handwriting recognition} | |
| 76 | |
| 77 | |
| 78 | |
| 79 \section{Introduction} | |
| 80 %\vspace*{-1mm} | |
| 81 | |
| 82 {\bf Deep Learning} has emerged as a promising new area of research in | |
| 83 statistical machine learning~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,VincentPLarochelleH2008-very-small,ranzato-08,TaylorHintonICML2009,Larochelle-jmlr-2009,Salakhutdinov+Hinton-2009,HonglakL2009,HonglakLNIPS2009,Jarrett-ICCV2009,Taylor-cvpr-2010}. See \citet{Bengio-2009} for a review. | |
| 84 Learning algorithms for deep architectures are centered on the learning | |
| 85 of useful representations of data, which are better suited to the task at hand, | |
| 86 and are organized in a hierarchy with multiple levels. | |
| 87 This is in part inspired by observations of the mammalian visual cortex, | |
| 88 which consists of a chain of processing elements, each of which is associated with a | |
| 89 different representation of the raw visual input. In fact, | |
| 90 it was found recently that the features learnt in deep architectures resemble | |
| 91 those observed in the first two of these stages (in areas V1 and V2 | |
| 92 of visual cortex) \citep{HonglakL2008}, and that they become more and | |
| 93 more invariant to factors of variation (such as camera movement) in | |
| 94 higher layers~\citep{Goodfellow2009}. | |
| 95 It has been hypothesized that learning a hierarchy of features increases the | |
| 96 ease and practicality of developing representations that are at once | |
| 97 tailored to specific tasks, yet are able to borrow statistical strength | |
| 98 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the | |
| 99 feature representation can lead to higher-level (more abstract, more | |
| 100 general) features that are more robust to unanticipated sources of | |
| 101 variance extant in real data. | |
| 102 | |
| 103 Whereas a deep architecture can in principle be more powerful than a | |
| 104 shallow one in terms of representation, depth appears to render the | |
| 105 training problem more difficult in terms of optimization and local minima. | |
| 106 It is also only recently that successful algorithms were proposed to | |
| 107 overcome some of these difficulties. All are based on unsupervised | |
| 108 learning, often in an greedy layer-wise ``unsupervised pre-training'' | |
| 109 stage~\citep{Bengio-2009}. | |
| 110 The principle is that each layer starting from | |
| 111 the bottom is trained to represent its input (the output of the previous | |
| 112 layer). After this | |
| 113 unsupervised initialization, the stack of layers can be | |
| 114 converted into a deep supervised feedforward neural network and fine-tuned by | |
| 115 stochastic gradient descent. | |
| 116 One of these layer initialization techniques, | |
| 117 applied here, is the Denoising | |
| 118 Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see | |
| 119 Figure~\ref{fig:da}), which performed similarly or | |
| 120 better~\citep{VincentPLarochelleH2008-very-small} than previously | |
| 121 proposed Restricted Boltzmann Machines (RBM)~\citep{Hinton06} | |
| 122 in terms of unsupervised extraction | |
| 123 of a hierarchy of features useful for classification. Each layer is trained | |
| 124 to denoise its input, creating a layer of features that can be used as | |
| 125 input for the next layer, forming a Stacked Denoising Auto-encoder (SDA). | |
| 126 Note that training a Denoising Auto-encoder | |
| 127 can actually been seen as training a particular RBM by an inductive | |
| 128 principle different from maximum likelihood~\citep{Vincent-SM-2010}, | |
| 129 namely by Score Matching~\citep{Hyvarinen-2005,HyvarinenA2008}. | |
| 130 | |
| 131 Previous comparative experimental results with stacking of RBMs and DAs | |
| 132 to build deep supervised predictors had shown that they could outperform | |
| 133 shallow architectures in a variety of settings, especially | |
| 134 when the data involves complex interactions between many factors of | |
| 135 variation~\citep{LarochelleH2007,Bengio-2009}. Other experiments have suggested | |
| 136 that the unsupervised layer-wise pre-training acted as a useful | |
| 137 prior~\citep{Erhan+al-2010} that allows one to initialize a deep | |
| 138 neural network in a relatively much smaller region of parameter space, | |
| 139 corresponding to better generalization. | |
| 140 | |
| 141 To further the understanding of the reasons for the good performance | |
| 142 observed with deep learners, we focus here on the following {\em hypothesis}: | |
| 143 intermediate levels of representation, especially when there are | |
| 144 more such levels, can be exploited to {\bf share | |
| 145 statistical strength across different but related types of examples}, | |
| 146 such as examples coming from other tasks than the task of interest | |
| 639 | 147 (the multi-task setting~\citep{caruana97a}), or examples coming from an overlapping |
| 627 | 148 but different distribution (images with different kinds of perturbations |
| 149 and noises, here). This is consistent with the hypotheses discussed | |
| 150 in~\citet{Bengio-2009} regarding the potential advantage | |
| 151 of deep learning and the idea that more levels of representation can | |
| 152 give rise to more abstract, more general features of the raw input. | |
| 153 | |
| 154 This hypothesis is related to a learning setting called | |
| 155 {\bf self-taught learning}~\citep{RainaR2007}, which combines principles | |
| 639 | 156 of semi-supervised and multi-task learning: in addition to the labeled |
| 157 examples from the target distribution, the learner can exploit examples | |
| 627 | 158 that are unlabeled and possibly come from a distribution different from the target |
| 159 distribution, e.g., from other classes than those of interest. | |
| 160 It has already been shown that deep learners can clearly take advantage of | |
| 639 | 161 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small} |
| 162 in order to improve performance on a supervised task, | |
| 627 | 163 but more needed to be done to explore the impact |
| 164 of {\em out-of-distribution} examples and of the {\em multi-task} setting | |
| 639 | 165 (two exceptions are~\citet{CollobertR2008}, which shares and uses unsupervised |
| 166 pre-training only with the first layer, and~\citet{icml2009_093} in the case | |
| 167 of video data). In particular the {\em relative | |
| 627 | 168 advantage of deep learning} for these settings has not been evaluated. |
| 169 | |
| 170 | |
| 171 % | |
| 172 The {\bf main claim} of this paper is that deep learners (with several levels of representation) can | |
| 173 {\bf benefit more from out-of-distribution examples than shallow learners} (with a single | |
| 174 level), both in the context of the multi-task setting and from | |
| 175 perturbed examples. Because we are able to improve on state-of-the-art | |
| 176 performance and reach human-level performance | |
| 177 on a large-scale task, we consider that this paper is also a contribution | |
| 178 to advance the application of machine learning to handwritten character recognition. | |
| 179 More precisely, we ask and answer the following questions: | |
| 180 | |
| 181 %\begin{enumerate} | |
| 182 $\bullet$ %\item | |
| 183 Do the good results previously obtained with deep architectures on the | |
| 184 MNIST digit images generalize to the setting of a similar but much larger and richer | |
| 185 dataset, the NIST special database 19, with 62 classes and around 800k examples? | |
| 186 | |
| 187 $\bullet$ %\item | |
| 188 To what extent does the perturbation of input images (e.g. adding | |
| 189 noise, affine transformations, background images) make the resulting | |
| 190 classifiers better not only on similarly perturbed images but also on | |
| 191 the {\em original clean examples}? We study this question in the | |
| 192 context of the 62-class and 10-class tasks of the NIST special database 19. | |
| 193 | |
| 194 $\bullet$ %\item | |
| 195 Do deep architectures {\em benefit {\bf more} from such out-of-distribution} | |
| 196 examples, in particular do they benefit more from | |
| 197 examples that are perturbed versions of the examples from the task of interest? | |
| 198 | |
| 199 $\bullet$ %\item | |
| 200 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more} | |
| 201 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than | |
| 202 a corresponding shallow and purely supervised architecture? | |
| 203 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case) | |
| 204 to answer this question. | |
| 205 %\end{enumerate} | |
| 206 | |
| 207 Our experimental results provide positive evidence towards all of these questions, | |
| 208 as well as {\bf classifiers that reach human-level performance on 62-class isolated character | |
| 209 recognition and beat previously published results on the NIST dataset (special database 19)}. | |
| 210 To achieve these results, we introduce in the next section a sophisticated system | |
| 211 for stochastically transforming character images and then explain the methodology, | |
| 212 which is based on training with or without these transformed images and testing on | |
| 213 clean ones. | |
| 214 Code for generating these transformations as well as for the deep learning | |
| 634 | 215 algorithms are made available at |
| 216 {\tt http://hg.assembla.com/ift6266}. | |
| 627 | 217 |
| 218 %\vspace*{-3mm} | |
| 219 %\newpage | |
| 220 \section{Perturbed and Transformed Character Images} | |
| 221 \label{s:perturbations} | |
| 222 %\vspace*{-2mm} | |
| 223 | |
| 224 Figure~\ref{fig:transform} shows the different transformations we used to stochastically | |
| 225 transform $32 \times 32$ source images (such as the one in Fig.\ref{fig:torig}) | |
| 226 in order to obtain data from a larger distribution which | |
| 227 covers a domain substantially larger than the clean characters distribution from | |
| 228 which we start. | |
| 229 Although character transformations have been used before to | |
| 230 improve character recognizers, this effort is on a large scale both | |
| 231 in number of classes and in the complexity of the transformations, hence | |
| 232 in the complexity of the learning task. | |
| 233 The code for these transformations (mostly Python) is available at | |
| 634 | 234 {\tt http://hg.assembla.com/ift6266}. All the modules in the pipeline (Figure~\ref{fig:transform}) share |
| 627 | 235 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the |
| 236 amount of deformation or noise introduced. | |
| 237 There are two main parts in the pipeline. The first one, | |
| 238 from thickness to pinch, performs transformations. The second | |
| 239 part, from blur to contrast, adds different kinds of noise. | |
| 639 | 240 More details can be found in~\citet{ARXIV-2010}. |
| 627 | 241 |
| 242 \begin{figure*}[ht] | |
| 243 \centering | |
| 244 \subfigure[Original]{\includegraphics[scale=0.6]{images/Original.png}\label{fig:torig}} | |
| 245 \subfigure[Thickness]{\includegraphics[scale=0.6]{images/Thick_only.png}} | |
| 246 \subfigure[Slant]{\includegraphics[scale=0.6]{images/Slant_only.png}} | |
| 247 \subfigure[Affine Transformation]{\includegraphics[scale=0.6]{images/Affine_only.png}} | |
| 248 \subfigure[Local Elastic Deformation]{\includegraphics[scale=0.6]{images/Localelasticdistorsions_only.png}} | |
| 249 \subfigure[Pinch]{\includegraphics[scale=0.6]{images/Pinch_only.png}} | |
| 250 %Noise | |
| 251 \subfigure[Motion Blur]{\includegraphics[scale=0.6]{images/Motionblur_only.png}} | |
| 252 \subfigure[Occlusion]{\includegraphics[scale=0.6]{images/occlusion_only.png}} | |
| 253 \subfigure[Gaussian Smoothing]{\includegraphics[scale=0.6]{images/Bruitgauss_only.png}} | |
| 254 \subfigure[Pixels Permutation]{\includegraphics[scale=0.6]{images/Permutpixel_only.png}} | |
| 255 \subfigure[Gaussian Noise]{\includegraphics[scale=0.6]{images/Distorsiongauss_only.png}} | |
| 256 \subfigure[Background Image Addition]{\includegraphics[scale=0.6]{images/background_other_only.png}} | |
| 257 \subfigure[Salt \& Pepper]{\includegraphics[scale=0.6]{images/Poivresel_only.png}} | |
| 258 \subfigure[Scratches]{\includegraphics[scale=0.6]{images/Rature_only.png}} | |
| 259 \subfigure[Grey Level \& Contrast]{\includegraphics[scale=0.6]{images/Contrast_only.png}} | |
| 260 \caption{Top left (a): example original image. Others (b-o): examples of the effect | |
| 261 of each transformation module taken separately. Actual perturbed examples are obtained by | |
| 262 a pipeline of these, with random choices about which module to apply and how much perturbation | |
| 263 to apply.} | |
| 264 \label{fig:transform} | |
| 265 %\vspace*{-2mm} | |
| 266 \end{figure*} | |
| 267 | |
| 268 %\vspace*{-3mm} | |
| 269 \section{Experimental Setup} | |
| 270 %\vspace*{-1mm} | |
| 271 | |
| 272 Much previous work on deep learning had been performed on | |
| 273 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009}, | |
| 274 with 60,000 examples, and variants involving 10,000 | |
| 639 | 275 examples~\citep{Larochelle-jmlr-2009,VincentPLarochelleH2008-very-small}\footnote{Fortunately, there |
| 276 are more and more exceptions of course, such as~\citet{RainaICML09} using a million examples.} | |
| 627 | 277 The focus here is on much larger training sets, from 10 times to |
| 278 to 1000 times larger, and 62 classes. | |
| 279 | |
| 280 The first step in constructing the larger datasets (called NISTP and P07) is to sample from | |
| 281 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas}, | |
| 282 and {\bf OCR data} (scanned machine printed characters). See more in | |
| 283 Section~\ref{sec:sources} below. Once a character | |
| 284 is sampled from one of these sources (chosen randomly), the second step is to | |
| 285 apply a pipeline of transformations and/or noise processes outlined in section \ref{s:perturbations}. | |
| 286 | |
| 287 To provide a baseline of error rate comparison we also estimate human performance | |
| 288 on both the 62-class task and the 10-class digits task. | |
| 289 We compare the best Multi-Layer Perceptrons (MLP) against | |
| 290 the best Stacked Denoising Auto-encoders (SDA), when | |
| 291 both models' hyper-parameters are selected to minimize the validation set error. | |
| 292 We also provide a comparison against a precise estimate | |
| 293 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 294 service ({\tt http://mturk.com}). | |
| 295 AMT users are paid small amounts | |
| 296 of money to perform tasks for which human intelligence is required. | |
| 297 Mechanical Turk has been used extensively in natural language processing and vision. | |
| 298 %processing \citep{SnowEtAl2008} and vision | |
| 299 %\citep{SorokinAndForsyth2008,whitehill09}. | |
| 300 AMT users were presented | |
| 301 with 10 character images (from a test set) on a screen | |
| 302 and asked to label them. | |
| 303 They were forced to choose a single character class (either among the | |
| 304 62 or 10 character classes) for each image. | |
| 305 80 subjects classified 2500 images per (dataset,task) pair. | |
| 306 Different humans labelers sometimes provided a different label for the same | |
| 307 example, and we were able to estimate the error variance due to this effect | |
| 308 because each image was classified by 3 different persons. | |
| 309 The average error of humans on the 62-class task NIST test set | |
| 310 is 18.2\%, with a standard error of 0.1\%. | |
| 311 We controlled noise in the labelling process by (1) | |
| 312 requiring AMT workers with a higher than normal average of accepted | |
| 313 responses ($>$95\%) on other tasks (2) discarding responses that were not | |
| 314 complete (10 predictions) (3) discarding responses for which for which the | |
| 315 time to predict was smaller than 3 seconds for NIST (the mean response time | |
| 316 was 20 seconds) and 6 seconds seconds for NISTP (average response time of | |
| 317 45 seconds) (4) discarding responses which were obviously wrong (10 | |
| 318 identical ones, or "12345..."). Overall, after such filtering, we kept | |
| 319 approximately 95\% of the AMT workers' responses. | |
| 320 | |
| 321 %\vspace*{-3mm} | |
| 322 \subsection{Data Sources} | |
| 323 \label{sec:sources} | |
| 324 %\vspace*{-2mm} | |
| 325 | |
| 326 %\begin{itemize} | |
| 327 %\item | |
| 328 {\bf NIST.} | |
| 329 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995}, | |
| 330 widely used for training and testing character | |
| 331 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}. | |
| 332 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications, | |
| 333 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes | |
| 334 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity. | |
| 335 The fourth partition (called $hsf_4$, 82,587 examples), | |
| 336 experimentally recognized to be the most difficult one, is the one recommended | |
| 337 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} | |
| 338 for that purpose. We randomly split the remainder (731,668 examples) into a training set and a validation set for | |
| 339 model selection. | |
| 340 The performances reported by previous work on that dataset mostly use only the digits. | |
| 341 Here we use all the classes both in the training and testing phase. This is especially | |
| 342 useful to estimate the effect of a multi-task setting. | |
| 343 The distribution of the classes in the NIST training and test sets differs | |
| 344 substantially, with relatively many more digits in the test set, and a more uniform distribution | |
| 345 of letters in the test set (whereas in the training set they are distributed | |
| 346 more like in natural text). | |
| 347 %\vspace*{-1mm} | |
| 348 | |
| 349 %\item | |
| 350 {\bf Fonts.} | |
| 351 In order to have a good variety of sources we downloaded an important number of free fonts from: | |
| 352 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}. | |
| 353 % TODO: pointless to anonymize, it's not pointing to our work | |
| 639 | 354 Including an operating system's (Windows 7) fonts, there we uniformly chose from $9817$ different fonts. |
| 627 | 355 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, |
| 356 directly as input to our models. | |
| 357 %\vspace*{-1mm} | |
| 358 | |
| 359 %\item | |
| 360 {\bf Captchas.} | |
| 361 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a Python-based captcha generator library) for | |
| 362 generating characters of the same format as the NIST dataset. This software is based on | |
| 363 a random character class generator and various kinds of transformations similar to those described in the previous sections. | |
| 364 In order to increase the variability of the data generated, many different fonts are used for generating the characters. | |
| 365 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity | |
| 366 depending on the value of the complexity parameter provided by the user of the data source. | |
| 367 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class? | |
| 368 %\vspace*{-1mm} | |
| 369 | |
| 370 %\item | |
| 371 {\bf OCR data.} | |
| 372 A large set (2 million) of scanned, OCRed and manually verified machine-printed | |
| 373 characters where included as an | |
| 374 additional source. This set is part of a larger corpus being collected by the Image Understanding | |
| 375 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern | |
| 639 | 376 ({\tt http://www.iupr.com}).%, and which will be publicly released. |
| 627 | 377 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this |
| 378 %\end{itemize} | |
| 379 | |
| 380 %\vspace*{-3mm} | |
| 381 \subsection{Data Sets} | |
| 382 %\vspace*{-2mm} | |
| 383 | |
| 639 | 384 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with one of 62 character labels. |
| 627 | 385 %\begin{itemize} |
| 386 %\vspace*{-1mm} | |
| 387 | |
| 388 %\item | |
| 389 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has | |
| 390 \{651,668 / 80,000 / 82,587\} \{training / validation / test\} examples. | |
| 391 %\vspace*{-1mm} | |
| 392 | |
| 393 %\item | |
| 639 | 394 {\bf P07.} This dataset is obtained by taking raw characters from the above 4 sources |
| 627 | 395 and sending them through the transformation pipeline described in section \ref{s:perturbations}. |
| 639 | 396 For each generated example, a data source is selected with probability $10\%$ from the fonts, |
| 397 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. The transformations are | |
| 398 applied in the | |
| 627 | 399 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$. |
| 400 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 401 obtained from the corresponding NIST sets plus other sources. | |
| 402 %\vspace*{-1mm} | |
| 403 | |
| 404 %\item | |
| 405 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources) | |
| 406 except that we only apply | |
| 407 transformations from slant to pinch (see Fig.\ref{fig:transform}(b-f)). | |
| 408 Therefore, the character is | |
| 639 | 409 transformed but without added noise, yielding images |
| 627 | 410 closer to the NIST dataset. |
| 411 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 412 obtained from the corresponding NIST sets plus other sources. | |
| 413 %\end{itemize} | |
| 414 | |
| 639 | 415 \vspace*{-3mm} |
| 627 | 416 \subsection{Models and their Hyper-parameters} |
| 417 %\vspace*{-2mm} | |
| 418 | |
| 419 The experiments are performed using MLPs (with a single | |
| 420 hidden layer) and deep SDAs. | |
| 421 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.} | |
| 422 | |
| 639 | 423 {\bf Multi-Layer Perceptrons (MLP).} The MLP output estimates the |
| 424 class-conditional probabilities | |
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425 \[ |
| 636 | 426 P({\rm class}|{\rm input}=x)={\rm softmax}(b_2+W_2\tanh(b_1+W_1 x)), |
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427 \] |
| 638 | 428 i.e., two layers, where $p={\rm softmax}(a)$ means that |
| 429 $p_i(x)=\exp(a_i)/\sum_j \exp(a_j)$ | |
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430 representing the probability |
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431 for class $i$, $\tanh$ is the element-wise |
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432 hyperbolic tangent, $b_i$ are parameter vectors, and $W_i$ are |
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433 parameter matrices (one per layer). The |
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434 number of rows of $W_1$ is called the number of hidden units (of the |
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435 single hidden layer, here), and |
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436 is one way to control capacity (the main other ways to control capacity are |
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437 the number of training iterations and optionally a regularization penalty |
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438 on the parameters, not used here because it did not help). |
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439 Whereas previous work had compared |
| 627 | 440 deep architectures to both shallow MLPs and SVMs, we only compared to MLPs |
| 441 here because of the very large datasets used (making the use of SVMs | |
| 442 computationally challenging because of their quadratic scaling | |
| 443 behavior). Preliminary experiments on training SVMs (libSVM) with subsets | |
| 444 of the training set allowing the program to fit in memory yielded | |
| 445 substantially worse results than those obtained with MLPs\footnote{RBF SVMs | |
| 446 trained with a subset of NISTP or NIST, 100k examples, to fit in memory, | |
| 447 yielded 64\% test error or worse; online linear SVMs trained on the whole | |
| 448 of NIST or 800k from NISTP yielded no better than 42\% error; slightly | |
| 449 better results were obtained by sparsifying the pixel intensities and | |
| 450 projecting to a second-order polynomial (a very sparse vector), still | |
| 451 41\% error. We expect that better results could be obtained with a | |
| 452 better implementation allowing for training with more examples and | |
| 453 a higher-order non-linear projection.} For training on nearly a hundred million examples (with the | |
| 454 perturbed data), the MLPs and SDA are much more convenient than classifiers | |
| 455 based on kernel methods. The MLP has a single hidden layer with $\tanh$ | |
| 456 activation functions, and softmax (normalized exponentials) on the output | |
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457 layer for estimating $P({\rm class} | {\rm input})$. The number of hidden units is |
| 627 | 458 taken in $\{300,500,800,1000,1500\}$. Training examples are presented in |
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459 minibatches of size 20, i.e., the parameters are iteratively updated in the direction |
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460 of the mean gradient of the next 20 examples. A constant learning rate was chosen among $\{0.001, |
| 627 | 461 0.01, 0.025, 0.075, 0.1, 0.5\}$. |
| 462 %through preliminary experiments (measuring performance on a validation set), | |
| 463 %and $0.1$ (which was found to work best) was then selected for optimizing on | |
| 464 %the whole training sets. | |
| 465 %\vspace*{-1mm} | |
| 466 | |
| 639 | 467 \begin{figure*}[htb] |
| 468 %\vspace*{-2mm} | |
| 469 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}} | |
| 470 %\vspace*{-2mm} | |
| 471 \caption{Illustration of the computations and training criterion for the denoising | |
| 472 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of | |
| 473 the layer (i.e. raw input or output of previous layer) | |
| 474 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$. | |
| 475 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which | |
| 476 is compared to the uncorrupted input $x$ through the loss function | |
| 477 $L_H(x,z)$, whose expected value is approximately minimized during training | |
| 478 by tuning $\theta$ and $\theta'$.} | |
| 479 \label{fig:da} | |
| 480 %\vspace*{-2mm} | |
| 481 \end{figure*} | |
| 482 | |
| 483 %\afterpage{\clearpage} | |
| 627 | 484 |
| 485 {\bf Stacked Denoising Auto-encoders (SDA).} | |
| 486 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs) | |
| 487 can be used to initialize the weights of each layer of a deep MLP (with many hidden | |
| 488 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006}, | |
| 489 apparently setting parameters in the | |
| 490 basin of attraction of supervised gradient descent yielding better | |
| 639 | 491 generalization~\citep{Erhan+al-2010}. |
| 492 This initial {\em unsupervised | |
| 493 pre-training phase} does not use the training labels. | |
| 627 | 494 Each layer is trained in turn to produce a new representation of its input |
| 495 (starting from the raw pixels). | |
| 496 It is hypothesized that the | |
| 497 advantage brought by this procedure stems from a better prior, | |
| 498 on the one hand taking advantage of the link between the input | |
| 499 distribution $P(x)$ and the conditional distribution of interest | |
| 500 $P(y|x)$ (like in semi-supervised learning), and on the other hand | |
| 501 taking advantage of the expressive power and bias implicit in the | |
| 502 deep architecture (whereby complex concepts are expressed as | |
| 503 compositions of simpler ones through a deep hierarchy). | |
| 504 | |
| 505 Here we chose to use the Denoising | |
| 506 Auto-encoder~\citep{VincentPLarochelleH2008-very-small} as the building block for | |
| 507 these deep hierarchies of features, as it is simple to train and | |
| 508 explain (see Figure~\ref{fig:da}, as well as | |
| 509 tutorial and code there: {\tt http://deeplearning.net/tutorial}), | |
| 510 provides efficient inference, and yielded results | |
| 511 comparable or better than RBMs in series of experiments | |
| 639 | 512 \citep{VincentPLarochelleH2008-very-small}. |
| 513 Some denoising auto-encoders correspond | |
| 514 to a Gaussian | |
| 627 | 515 RBM trained by a Score Matching criterion~\cite{Vincent-SM-2010}. |
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516 During its unsupervised training, a Denoising |
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517 Auto-encoder is presented with a stochastically corrupted version $\tilde{x}$ |
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518 of the input $x$ and trained to reconstruct to produce a reconstruction $z$ |
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519 of the uncorrupted input $x$. Because the network has to denoise, it is |
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520 forcing the hidden units $y$ to represent the leading regularities in |
| 639 | 521 the data. In a slight departure from \citet{VincentPLarochelleH2008-very-small}, |
| 522 the hidden units output $y$ is obtained through the tanh-affine | |
| 636 | 523 encoder |
| 639 | 524 $y=\tanh(c+V x)$ |
| 525 and the reconstruction is obtained through the transposed transformation | |
| 526 $z=\tanh(d+V' y)$. | |
| 637 | 527 The training |
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528 set average of the cross-entropy |
| 639 | 529 reconstruction loss (after mapping back numbers in (-1,1) into (0,1)) |
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530 \[ |
| 639 | 531 L_H(x,z)=-\sum_i \frac{(z_i+1)}{2} \log \frac{(x_i+1)}{2} + \frac{z_i}{2} \log\frac{x_i}{2} |
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532 \] |
| 637 | 533 is minimized. |
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534 Here we use the random binary masking corruption |
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535 (which in $\tilde{x}$ sets to 0 a random subset of the elements of $x$, and |
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536 copies the rest). |
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537 Once the first denoising auto-encoder is trained, its parameters can be used |
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538 to set the first layer of the deep MLP. The original data are then processed |
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539 through that first layer, and the output of the hidden units form a new |
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540 representation that can be used as input data for training a second denoising |
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541 auto-encoder, still in a purely unsupervised way. |
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542 This is repeated for the desired number of hidden layers. |
| 627 | 543 After this unsupervised pre-training stage, the parameters |
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544 are used to initialize a deep MLP (similar to the above, but |
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545 with more layers), which is fine-tuned by |
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546 the same standard procedure (stochastic gradient descent) |
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547 used to train MLPs in general (see above). |
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548 The top layer parameters of the deep MLP (the one which outputs the |
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549 class probabilities and takes the top hidden layer as input) can |
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550 be initialized at 0. |
| 627 | 551 The SDA hyper-parameters are the same as for the MLP, with the addition of the |
| 552 amount of corruption noise (we used the masking noise process, whereby a | |
| 553 fixed proportion of the input values, randomly selected, are zeroed), and a | |
| 554 separate learning rate for the unsupervised pre-training stage (selected | |
| 555 from the same above set). The fraction of inputs corrupted was selected | |
| 556 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number | |
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557 of hidden layers but it was fixed to 3 for our experiments, |
| 627 | 558 based on previous work with |
| 639 | 559 SDAs on MNIST~\citep{VincentPLarochelleH2008-very-small}. |
| 637 | 560 We also compared against 1 and against 2 hidden layers, |
| 561 to disantangle the effect of depth from that of unsupervised | |
| 627 | 562 pre-training. |
| 637 | 563 The size of each hidden |
| 564 layer was kept constant across hidden layers, and the best results | |
| 565 were obtained with the largest values that we tried | |
| 566 (1000 hidden units). | |
| 627 | 567 |
| 568 %\vspace*{-1mm} | |
| 569 | |
| 570 \begin{figure*}[ht] | |
| 571 %\vspace*{-2mm} | |
| 572 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}} | |
| 573 %\vspace*{-3mm} | |
| 574 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained | |
| 575 on NIST, 1 on NISTP, and 2 on P07. Left: overall results | |
| 576 of all models, on NIST and NISTP test sets. | |
| 577 Right: error rates on NIST test digits only, along with the previous results from | |
| 578 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005} | |
| 579 respectively based on ART, nearest neighbors, MLPs, and SVMs.} | |
| 580 \label{fig:error-rates-charts} | |
| 581 %\vspace*{-2mm} | |
| 582 \end{figure*} | |
| 583 | |
| 584 | |
| 585 \begin{figure*}[ht] | |
| 586 \vspace*{-3mm} | |
| 587 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}} | |
| 588 \vspace*{-3mm} | |
| 589 \caption{Relative improvement in error rate due to out-of-distribution examples. | |
| 590 Left: Improvement (or loss, when negative) | |
| 591 induced by out-of-distribution examples (perturbed data). | |
| 592 Right: Improvement (or loss, when negative) induced by multi-task | |
| 593 learning (training on all classes and testing only on either digits, | |
| 594 upper case, or lower-case). The deep learner (SDA) benefits more from | |
| 595 out-of-distribution examples, compared to the shallow MLP.} | |
| 596 \label{fig:improvements-charts} | |
| 597 \vspace*{-2mm} | |
| 598 \end{figure*} | |
| 599 | |
| 600 \vspace*{-2mm} | |
| 601 \section{Experimental Results} | |
| 602 \vspace*{-2mm} | |
| 603 | |
| 604 %%\vspace*{-1mm} | |
| 605 %\subsection{SDA vs MLP vs Humans} | |
| 606 %%\vspace*{-1mm} | |
| 607 The models are either trained on NIST (MLP0 and SDA0), | |
| 608 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested | |
| 609 on either NIST, NISTP or P07 (regardless of the data set used for training), | |
| 610 either on the 62-class task | |
| 611 or on the 10-digits task. Training time (including about half | |
| 612 for unsupervised pre-training, for DAs) on the larger | |
| 613 datasets is around one day on a GPU (GTX 285). | |
| 614 Figure~\ref{fig:error-rates-charts} summarizes the results obtained, | |
| 615 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1, | |
| 616 SDA2), along with the previous results on the digits NIST special database | |
| 617 19 test set from the literature, respectively based on ARTMAP neural | |
| 618 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search | |
| 619 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs | |
| 620 ~\citep{Milgram+al-2005}.% More detailed and complete numerical results | |
| 621 %(figures and tables, including standard errors on the error rates) can be | |
| 622 %found in Appendix. | |
| 623 The deep learner not only outperformed the shallow ones and | |
| 624 previously published performance (in a statistically and qualitatively | |
| 625 significant way) but when trained with perturbed data | |
| 626 reaches human performance on both the 62-class task | |
| 627 and the 10-class (digits) task. | |
| 628 17\% error (SDA1) or 18\% error (humans) may seem large but a large | |
| 629 majority of the errors from humans and from SDA1 are from out-of-context | |
| 630 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a | |
| 631 ``c'' and a ``C'' are often indistinguishible). | |
| 632 Regarding shallower networks pre-trained with unsupervised denoising | |
| 633 auto-encders, we find that the NIST test error is 21\% with one hidden | |
| 634 layer and 20\% with two hidden layers (vs 17\% in the same conditions | |
| 635 with 3 hidden layers). Compare this with the 23\% error achieved | |
| 636 by the MLP, i.e. a single hidden layer and no unsupervised pre-training. | |
| 637 As found in previous work~\cite{Erhan+al-2010,Larochelle-jmlr-2009}, | |
| 638 these results show that both depth and | |
| 639 unsupervised pre-training need to be combined in order to achieve | |
| 640 the best results. | |
| 641 | |
| 642 | |
| 643 In addition, as shown in the left of | |
| 644 Figure~\ref{fig:improvements-charts}, the relative improvement in error | |
| 645 rate brought by out-of-distribution examples is greater for the deep | |
| 646 SDA, and these | |
| 647 differences with the shallow MLP are statistically and qualitatively | |
| 648 significant. | |
| 649 The left side of the figure shows the improvement to the clean | |
| 650 NIST test set error brought by the use of out-of-distribution examples | |
| 651 (i.e. the perturbed examples examples from NISTP or P07), | |
| 652 over the models trained exclusively on NIST (respectively SDA0 and MLP0). | |
| 653 Relative percent change is measured by taking | |
| 654 $100 \% \times$ (original model's error / perturbed-data model's error - 1). | |
| 655 The right side of | |
| 656 Figure~\ref{fig:improvements-charts} shows the relative improvement | |
| 657 brought by the use of a multi-task setting, in which the same model is | |
| 658 trained for more classes than the target classes of interest (i.e. training | |
| 659 with all 62 classes when the target classes are respectively the digits, | |
| 660 lower-case, or upper-case characters). Again, whereas the gain from the | |
| 661 multi-task setting is marginal or negative for the MLP, it is substantial | |
| 662 for the SDA. Note that to simplify these multi-task experiments, only the original | |
| 663 NIST dataset is used. For example, the MLP-digits bar shows the relative | |
| 664 percent improvement in MLP error rate on the NIST digits test set | |
| 665 as $100\% \times$ (single-task | |
| 666 model's error / multi-task model's error - 1). The single-task model is | |
| 667 trained with only 10 outputs (one per digit), seeing only digit examples, | |
| 668 whereas the multi-task model is trained with 62 outputs, with all 62 | |
| 669 character classes as examples. Hence the hidden units are shared across | |
| 670 all tasks. For the multi-task model, the digit error rate is measured by | |
| 671 comparing the correct digit class with the output class associated with the | |
| 672 maximum conditional probability among only the digit classes outputs. The | |
| 673 setting is similar for the other two target classes (lower case characters | |
| 674 and upper case characters). Note however that some types of perturbations | |
| 675 (NISTP) help more than others (P07) when testing on the clean images. | |
| 676 %%\vspace*{-1mm} | |
| 677 %\subsection{Perturbed Training Data More Helpful for SDA} | |
| 678 %%\vspace*{-1mm} | |
| 679 | |
| 680 %%\vspace*{-1mm} | |
| 681 %\subsection{Multi-Task Learning Effects} | |
| 682 %%\vspace*{-1mm} | |
| 683 | |
| 684 \iffalse | |
| 685 As previously seen, the SDA is better able to benefit from the | |
| 686 transformations applied to the data than the MLP. In this experiment we | |
| 687 define three tasks: recognizing digits (knowing that the input is a digit), | |
| 688 recognizing upper case characters (knowing that the input is one), and | |
| 689 recognizing lower case characters (knowing that the input is one). We | |
| 690 consider the digit classification task as the target task and we want to | |
| 691 evaluate whether training with the other tasks can help or hurt, and | |
| 692 whether the effect is different for MLPs versus SDAs. The goal is to find | |
| 693 out if deep learning can benefit more (or less) from multiple related tasks | |
| 694 (i.e. the multi-task setting) compared to a corresponding purely supervised | |
| 695 shallow learner. | |
| 696 | |
| 697 We use a single hidden layer MLP with 1000 hidden units, and a SDA | |
| 698 with 3 hidden layers (1000 hidden units per layer), pre-trained and | |
| 699 fine-tuned on NIST. | |
| 700 | |
| 701 Our results show that the MLP benefits marginally from the multi-task setting | |
| 702 in the case of digits (5\% relative improvement) but is actually hurt in the case | |
| 703 of characters (respectively 3\% and 4\% worse for lower and upper class characters). | |
| 704 On the other hand the SDA benefited from the multi-task setting, with relative | |
| 705 error rate improvements of 27\%, 15\% and 13\% respectively for digits, | |
| 706 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. | |
| 707 \fi | |
| 708 | |
| 709 | |
| 710 \vspace*{-2mm} | |
| 711 \section{Conclusions and Discussion} | |
| 712 \vspace*{-2mm} | |
| 713 | |
| 714 We have found that out-of-distribution examples (multi-task learning | |
| 715 and perturbed examples) are more beneficial | |
| 716 to a deep learner than to a traditional shallow and purely | |
| 717 supervised learner. More precisely, | |
| 718 the answers are positive for all the questions asked in the introduction. | |
| 719 %\begin{itemize} | |
| 720 | |
| 721 $\bullet$ %\item | |
| 722 {\bf Do the good results previously obtained with deep architectures on the | |
| 723 MNIST digits generalize to a much larger and richer (but similar) | |
| 724 dataset, the NIST special database 19, with 62 classes and around 800k examples}? | |
| 725 Yes, the SDA {\em systematically outperformed the MLP and all the previously | |
| 726 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level | |
| 727 performance} at around 17\% error on the 62-class task and 1.4\% on the digits, | |
| 728 and beating previously published results on the same data. | |
| 729 | |
| 730 $\bullet$ %\item | |
| 731 {\bf To what extent do out-of-distribution examples help deep learners, | |
| 732 and do they help them more than shallow supervised ones}? | |
| 733 We found that distorted training examples not only made the resulting | |
| 734 classifier better on similarly perturbed images but also on | |
| 735 the {\em original clean examples}, and more importantly and more novel, | |
| 736 that deep architectures benefit more from such {\em out-of-distribution} | |
| 737 examples. Shallow MLPs were helped by perturbed training examples when tested on perturbed input | |
| 738 images (65\% relative improvement on NISTP) | |
| 739 but only marginally helped (5\% relative improvement on all classes) | |
| 740 or even hurt (10\% relative loss on digits) | |
| 741 with respect to clean examples. On the other hand, the deep SDAs | |
| 742 were significantly boosted by these out-of-distribution examples. | |
| 743 Similarly, whereas the improvement due to the multi-task setting was marginal or | |
| 744 negative for the MLP (from +5.6\% to -3.6\% relative change), | |
| 745 it was quite significant for the SDA (from +13\% to +27\% relative change), | |
| 746 which may be explained by the arguments below. | |
| 747 Since out-of-distribution data | |
| 748 (perturbed or from other related classes) is very common, this conclusion | |
| 749 is of practical importance. | |
| 750 %\end{itemize} | |
| 751 | |
| 752 In the original self-taught learning framework~\citep{RainaR2007}, the | |
| 753 out-of-sample examples were used as a source of unsupervised data, and | |
| 754 experiments showed its positive effects in a \emph{limited labeled data} | |
| 755 scenario. However, many of the results by \citet{RainaR2007} (who used a | |
| 756 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught | |
| 757 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases. | |
| 758 We note instead that, for deep | |
| 759 architectures, our experiments show that such a positive effect is accomplished | |
| 760 even in a scenario with a \emph{large number of labeled examples}, | |
| 761 i.e., here, the relative gain of self-taught learning and | |
| 762 out-of-distribution examples is probably preserved | |
| 763 in the asymptotic regime. However, note that in our perturbation experiments | |
| 764 (but not in our multi-task experiments), | |
| 765 even the out-of-distribution examples are labeled, unlike in the | |
| 766 earlier self-taught learning experiments~\citep{RainaR2007}. | |
| 767 | |
| 768 {\bf Why would deep learners benefit more from the self-taught learning | |
| 769 framework and out-of-distribution examples}? | |
| 770 The key idea is that the lower layers of the predictor compute a hierarchy | |
| 771 of features that can be shared across tasks or across variants of the | |
| 772 input distribution. A theoretical analysis of generalization improvements | |
| 773 due to sharing of intermediate features across tasks already points | |
| 639 | 774 towards that explanation~\citep{baxter95a}. |
| 627 | 775 Intermediate features that can be used in different |
| 776 contexts can be estimated in a way that allows to share statistical | |
| 777 strength. Features extracted through many levels are more likely to | |
| 778 be more abstract and more invariant to some of the factors of variation | |
| 779 in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest), | |
| 780 increasing the likelihood that they would be useful for a larger array | |
| 781 of tasks and input conditions. | |
| 782 Therefore, we hypothesize that both depth and unsupervised | |
| 783 pre-training play a part in explaining the advantages observed here, and future | |
| 784 experiments could attempt at teasing apart these factors. | |
| 785 And why would deep learners benefit from the self-taught learning | |
| 786 scenarios even when the number of labeled examples is very large? | |
| 787 We hypothesize that this is related to the hypotheses studied | |
| 788 in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010} | |
| 789 it was found that online learning on a huge dataset did not make the | |
| 790 advantage of the deep learning bias vanish, and a similar phenomenon | |
| 791 may be happening here. We hypothesize that unsupervised pre-training | |
| 792 of a deep hierarchy with out-of-distribution examples initializes the | |
| 793 model in the basin of attraction of supervised gradient descent | |
| 794 that corresponds to better generalization. Furthermore, such good | |
| 795 basins of attraction are not discovered by pure supervised learning | |
| 796 (with or without out-of-distribution examples) from random initialization, and more labeled examples | |
| 797 does not allow the shallow or purely supervised models to discover | |
| 798 the kind of better basins associated | |
| 799 with deep learning and out-of-distribution examples. | |
| 800 | |
| 639 | 801 A Java demo of the recognizer (where both the MLP and the SDA can be compared) |
| 634 | 802 can be executed on-line at {\tt http://deep.host22.com}. |
| 627 | 803 |
| 804 \iffalse | |
| 805 \section*{Appendix I: Detailed Numerical Results} | |
| 806 | |
| 807 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered. | |
| 808 They also contain additional data such as test errors on P07 and standard errors. | |
| 809 | |
| 810 \begin{table}[ht] | |
| 811 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + | |
| 812 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training | |
| 813 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture | |
| 814 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) | |
| 815 and using a validation set to select hyper-parameters and other training choices. | |
| 816 \{SDA,MLP\}0 are trained on NIST, | |
| 817 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. | |
| 818 The human error rate on digits is a lower bound because it does not count digits that were | |
| 819 recognized as letters. For comparison, the results found in the literature | |
| 820 on NIST digits classification using the same test set are included.} | |
| 821 \label{tab:sda-vs-mlp-vs-humans} | |
| 822 \begin{center} | |
| 823 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 824 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 825 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline | |
| 826 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline | |
| 827 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline | |
| 828 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline | |
| 829 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline | |
| 830 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline | |
| 831 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline | |
| 832 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline | |
| 833 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline | |
| 834 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline | |
| 835 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline | |
| 836 \end{tabular} | |
| 837 \end{center} | |
| 838 \end{table} | |
| 839 | |
| 840 \begin{table}[ht] | |
| 841 \caption{Relative change in error rates due to the use of perturbed training data, | |
| 842 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. | |
| 843 A positive value indicates that training on the perturbed data helped for the | |
| 844 given test set (the first 3 columns on the 62-class tasks and the last one is | |
| 845 on the clean 10-class digits). Clearly, the deep learning models did benefit more | |
| 846 from perturbed training data, even when testing on clean data, whereas the MLP | |
| 847 trained on perturbed data performed worse on the clean digits and about the same | |
| 848 on the clean characters. } | |
| 849 \label{tab:perturbation-effect} | |
| 850 \begin{center} | |
| 851 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 852 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 853 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline | |
| 854 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline | |
| 855 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline | |
| 856 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline | |
| 857 \end{tabular} | |
| 858 \end{center} | |
| 859 \end{table} | |
| 860 | |
| 861 \begin{table}[ht] | |
| 862 \caption{Test error rates and relative change in error rates due to the use of | |
| 863 a multi-task setting, i.e., training on each task in isolation vs training | |
| 864 for all three tasks together, for MLPs vs SDAs. The SDA benefits much | |
| 865 more from the multi-task setting. All experiments on only on the | |
| 866 unperturbed NIST data, using validation error for model selection. | |
| 867 Relative improvement is 1 - single-task error / multi-task error.} | |
| 868 \label{tab:multi-task} | |
| 869 \begin{center} | |
| 870 \begin{tabular}{|l|r|r|r|} \hline | |
| 871 & single-task & multi-task & relative \\ | |
| 872 & setting & setting & improvement \\ \hline | |
| 873 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline | |
| 874 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline | |
| 875 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline | |
| 876 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline | |
| 877 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline | |
| 878 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline | |
| 879 \end{tabular} | |
| 880 \end{center} | |
| 881 \end{table} | |
| 882 | |
| 883 \fi | |
| 884 | |
| 885 %\afterpage{\clearpage} | |
| 886 %\clearpage | |
| 887 { | |
| 888 %\bibliographystyle{spbasic} % basic style, author-year citations | |
| 889 \bibliographystyle{plainnat} | |
| 890 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter} | |
| 891 %\bibliographystyle{unsrtnat} | |
| 892 %\bibliographystyle{apalike} | |
| 893 } | |
| 894 | |
| 895 | |
| 896 \end{document} |