Mercurial > ift6266
annotate writeup/aistats2011_cameraready.tex @ 630:f55f1b1499c4
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| author | Frederic Bastien <nouiz@nouiz.org> |
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| date | Thu, 17 Mar 2011 15:51:43 -0400 |
| parents | 249a180795e3 |
| children | 510220effb14 |
| 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} | |
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| 20 | |
| 21 \begin{document} | |
| 22 | |
| 23 \twocolumn[ | |
| 24 \aistatstitle{Deep Learners Benefit More from Out-of-Distribution Examples} | |
| 25 \runningtitle{Deep Learners for Out-of-Distribution Examples} | |
| 26 \runningauthor{Bengio et. al.} | |
| 27 \aistatsauthor{ | |
| 28 Yoshua Bengio \and | |
| 29 Frédéric Bastien \and | |
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30 \bf Arnaud Bergeron \and |
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31 Nicolas Boulanger-Lewandowski \and \\ |
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32 \bf Thomas Breuel \and |
| 627 | 33 Youssouf Chherawala \and |
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34 \bf Moustapha Cisse \and |
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35 Myriam Côté \and \\ |
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36 \bf Dumitru Erhan \and |
| 627 | 37 Jeremy Eustache \and |
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38 \bf Xavier Glorot \and |
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39 Xavier Muller \and \\ |
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40 \bf Sylvain Pannetier Lebeuf \and |
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41 Razvan Pascanu \and |
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42 \bf Salah Rifai \and |
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43 Francois Savard \and \\ |
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44 \bf Guillaume Sicard \\ |
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45 \vspace*{1mm}} |
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46 |
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47 %I can't use aistatsaddress in a single side paragraphe. |
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48 %The document is 2 colums, but this section span the 2 colums, sot there is only 1 left |
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49 \center{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} |
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50 \vspace*{5mm} |
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51 ] |
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52 %\aistatsaddress{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} |
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53 |
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54 |
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55 %\vspace*{5mm}} |
| 627 | 56 %\date{{\tt bengioy@iro.umontreal.ca}, Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} |
| 57 %\jmlrheading{}{2010}{}{10/2010}{XX/2011}{Yoshua Bengio et al} | |
| 58 %\editor{} | |
| 59 | |
| 60 %\makeanontitle | |
| 61 %\maketitle | |
| 62 | |
| 63 %{\bf Running title: Deep Self-Taught Learning} | |
| 64 | |
| 65 \vspace*{5mm} | |
| 66 \begin{abstract} | |
| 67 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}. | |
| 68 \end{abstract} | |
| 69 %\vspace*{-3mm} | |
| 70 | |
| 71 %\begin{keywords} | |
| 72 %Deep learning, self-taught learning, out-of-distribution examples, handwritten character recognition, multi-task learning | |
| 73 %\end{keywords} | |
| 74 %\keywords{self-taught learning \and multi-task learning \and out-of-distribution examples \and deep learning \and handwriting recognition} | |
| 75 | |
| 76 | |
| 77 | |
| 78 \section{Introduction} | |
| 79 %\vspace*{-1mm} | |
| 80 | |
| 81 {\bf Deep Learning} has emerged as a promising new area of research in | |
| 82 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. | |
| 83 Learning algorithms for deep architectures are centered on the learning | |
| 84 of useful representations of data, which are better suited to the task at hand, | |
| 85 and are organized in a hierarchy with multiple levels. | |
| 86 This is in part inspired by observations of the mammalian visual cortex, | |
| 87 which consists of a chain of processing elements, each of which is associated with a | |
| 88 different representation of the raw visual input. In fact, | |
| 89 it was found recently that the features learnt in deep architectures resemble | |
| 90 those observed in the first two of these stages (in areas V1 and V2 | |
| 91 of visual cortex) \citep{HonglakL2008}, and that they become more and | |
| 92 more invariant to factors of variation (such as camera movement) in | |
| 93 higher layers~\citep{Goodfellow2009}. | |
| 94 It has been hypothesized that learning a hierarchy of features increases the | |
| 95 ease and practicality of developing representations that are at once | |
| 96 tailored to specific tasks, yet are able to borrow statistical strength | |
| 97 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the | |
| 98 feature representation can lead to higher-level (more abstract, more | |
| 99 general) features that are more robust to unanticipated sources of | |
| 100 variance extant in real data. | |
| 101 | |
| 102 Whereas a deep architecture can in principle be more powerful than a | |
| 103 shallow one in terms of representation, depth appears to render the | |
| 104 training problem more difficult in terms of optimization and local minima. | |
| 105 It is also only recently that successful algorithms were proposed to | |
| 106 overcome some of these difficulties. All are based on unsupervised | |
| 107 learning, often in an greedy layer-wise ``unsupervised pre-training'' | |
| 108 stage~\citep{Bengio-2009}. | |
| 109 The principle is that each layer starting from | |
| 110 the bottom is trained to represent its input (the output of the previous | |
| 111 layer). After this | |
| 112 unsupervised initialization, the stack of layers can be | |
| 113 converted into a deep supervised feedforward neural network and fine-tuned by | |
| 114 stochastic gradient descent. | |
| 115 One of these layer initialization techniques, | |
| 116 applied here, is the Denoising | |
| 117 Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see | |
| 118 Figure~\ref{fig:da}), which performed similarly or | |
| 119 better~\citep{VincentPLarochelleH2008-very-small} than previously | |
| 120 proposed Restricted Boltzmann Machines (RBM)~\citep{Hinton06} | |
| 121 in terms of unsupervised extraction | |
| 122 of a hierarchy of features useful for classification. Each layer is trained | |
| 123 to denoise its input, creating a layer of features that can be used as | |
| 124 input for the next layer, forming a Stacked Denoising Auto-encoder (SDA). | |
| 125 Note that training a Denoising Auto-encoder | |
| 126 can actually been seen as training a particular RBM by an inductive | |
| 127 principle different from maximum likelihood~\citep{Vincent-SM-2010}, | |
| 128 namely by Score Matching~\citep{Hyvarinen-2005,HyvarinenA2008}. | |
| 129 | |
| 130 Previous comparative experimental results with stacking of RBMs and DAs | |
| 131 to build deep supervised predictors had shown that they could outperform | |
| 132 shallow architectures in a variety of settings, especially | |
| 133 when the data involves complex interactions between many factors of | |
| 134 variation~\citep{LarochelleH2007,Bengio-2009}. Other experiments have suggested | |
| 135 that the unsupervised layer-wise pre-training acted as a useful | |
| 136 prior~\citep{Erhan+al-2010} that allows one to initialize a deep | |
| 137 neural network in a relatively much smaller region of parameter space, | |
| 138 corresponding to better generalization. | |
| 139 | |
| 140 To further the understanding of the reasons for the good performance | |
| 141 observed with deep learners, we focus here on the following {\em hypothesis}: | |
| 142 intermediate levels of representation, especially when there are | |
| 143 more such levels, can be exploited to {\bf share | |
| 144 statistical strength across different but related types of examples}, | |
| 145 such as examples coming from other tasks than the task of interest | |
| 146 (the multi-task setting), or examples coming from an overlapping | |
| 147 but different distribution (images with different kinds of perturbations | |
| 148 and noises, here). This is consistent with the hypotheses discussed | |
| 149 in~\citet{Bengio-2009} regarding the potential advantage | |
| 150 of deep learning and the idea that more levels of representation can | |
| 151 give rise to more abstract, more general features of the raw input. | |
| 152 | |
| 153 This hypothesis is related to a learning setting called | |
| 154 {\bf self-taught learning}~\citep{RainaR2007}, which combines principles | |
| 155 of semi-supervised and multi-task learning: the learner can exploit examples | |
| 156 that are unlabeled and possibly come from a distribution different from the target | |
| 157 distribution, e.g., from other classes than those of interest. | |
| 158 It has already been shown that deep learners can clearly take advantage of | |
| 159 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small}, | |
| 160 but more needed to be done to explore the impact | |
| 161 of {\em out-of-distribution} examples and of the {\em multi-task} setting | |
| 162 (one exception is~\citep{CollobertR2008}, which shares and uses unsupervised | |
| 163 pre-training only with the first layer). In particular the {\em relative | |
| 164 advantage of deep learning} for these settings has not been evaluated. | |
| 165 | |
| 166 | |
| 167 % | |
| 168 The {\bf main claim} of this paper is that deep learners (with several levels of representation) can | |
| 169 {\bf benefit more from out-of-distribution examples than shallow learners} (with a single | |
| 170 level), both in the context of the multi-task setting and from | |
| 171 perturbed examples. Because we are able to improve on state-of-the-art | |
| 172 performance and reach human-level performance | |
| 173 on a large-scale task, we consider that this paper is also a contribution | |
| 174 to advance the application of machine learning to handwritten character recognition. | |
| 175 More precisely, we ask and answer the following questions: | |
| 176 | |
| 177 %\begin{enumerate} | |
| 178 $\bullet$ %\item | |
| 179 Do the good results previously obtained with deep architectures on the | |
| 180 MNIST digit images generalize to the setting of a similar but much larger and richer | |
| 181 dataset, the NIST special database 19, with 62 classes and around 800k examples? | |
| 182 | |
| 183 $\bullet$ %\item | |
| 184 To what extent does the perturbation of input images (e.g. adding | |
| 185 noise, affine transformations, background images) make the resulting | |
| 186 classifiers better not only on similarly perturbed images but also on | |
| 187 the {\em original clean examples}? We study this question in the | |
| 188 context of the 62-class and 10-class tasks of the NIST special database 19. | |
| 189 | |
| 190 $\bullet$ %\item | |
| 191 Do deep architectures {\em benefit {\bf more} from such out-of-distribution} | |
| 192 examples, in particular do they benefit more from | |
| 193 examples that are perturbed versions of the examples from the task of interest? | |
| 194 | |
| 195 $\bullet$ %\item | |
| 196 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more} | |
| 197 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than | |
| 198 a corresponding shallow and purely supervised architecture? | |
| 199 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case) | |
| 200 to answer this question. | |
| 201 %\end{enumerate} | |
| 202 | |
| 203 Our experimental results provide positive evidence towards all of these questions, | |
| 204 as well as {\bf classifiers that reach human-level performance on 62-class isolated character | |
| 205 recognition and beat previously published results on the NIST dataset (special database 19)}. | |
| 206 To achieve these results, we introduce in the next section a sophisticated system | |
| 207 for stochastically transforming character images and then explain the methodology, | |
| 208 which is based on training with or without these transformed images and testing on | |
| 209 clean ones. | |
| 210 Code for generating these transformations as well as for the deep learning | |
| 211 algorithms are made available at {\tt http://anonymous.url.net}.%{\tt http://hg.assembla.com/ift6266}. | |
| 212 | |
| 213 %\vspace*{-3mm} | |
| 214 %\newpage | |
| 215 \section{Perturbed and Transformed Character Images} | |
| 216 \label{s:perturbations} | |
| 217 %\vspace*{-2mm} | |
| 218 | |
| 219 Figure~\ref{fig:transform} shows the different transformations we used to stochastically | |
| 220 transform $32 \times 32$ source images (such as the one in Fig.\ref{fig:torig}) | |
| 221 in order to obtain data from a larger distribution which | |
| 222 covers a domain substantially larger than the clean characters distribution from | |
| 223 which we start. | |
| 224 Although character transformations have been used before to | |
| 225 improve character recognizers, this effort is on a large scale both | |
| 226 in number of classes and in the complexity of the transformations, hence | |
| 227 in the complexity of the learning task. | |
| 228 The code for these transformations (mostly Python) is available at | |
| 229 {\tt http://anonymous.url.net}. All the modules in the pipeline (Figure~\ref{fig:transform}) share | |
| 230 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the | |
| 231 amount of deformation or noise introduced. | |
| 232 There are two main parts in the pipeline. The first one, | |
| 233 from thickness to pinch, performs transformations. The second | |
| 234 part, from blur to contrast, adds different kinds of noise. | |
| 235 More details can be found in~\citep{ift6266-tr-anonymous}. | |
| 236 | |
| 237 \begin{figure*}[ht] | |
| 238 \centering | |
| 239 \subfigure[Original]{\includegraphics[scale=0.6]{images/Original.png}\label{fig:torig}} | |
| 240 \subfigure[Thickness]{\includegraphics[scale=0.6]{images/Thick_only.png}} | |
| 241 \subfigure[Slant]{\includegraphics[scale=0.6]{images/Slant_only.png}} | |
| 242 \subfigure[Affine Transformation]{\includegraphics[scale=0.6]{images/Affine_only.png}} | |
| 243 \subfigure[Local Elastic Deformation]{\includegraphics[scale=0.6]{images/Localelasticdistorsions_only.png}} | |
| 244 \subfigure[Pinch]{\includegraphics[scale=0.6]{images/Pinch_only.png}} | |
| 245 %Noise | |
| 246 \subfigure[Motion Blur]{\includegraphics[scale=0.6]{images/Motionblur_only.png}} | |
| 247 \subfigure[Occlusion]{\includegraphics[scale=0.6]{images/occlusion_only.png}} | |
| 248 \subfigure[Gaussian Smoothing]{\includegraphics[scale=0.6]{images/Bruitgauss_only.png}} | |
| 249 \subfigure[Pixels Permutation]{\includegraphics[scale=0.6]{images/Permutpixel_only.png}} | |
| 250 \subfigure[Gaussian Noise]{\includegraphics[scale=0.6]{images/Distorsiongauss_only.png}} | |
| 251 \subfigure[Background Image Addition]{\includegraphics[scale=0.6]{images/background_other_only.png}} | |
| 252 \subfigure[Salt \& Pepper]{\includegraphics[scale=0.6]{images/Poivresel_only.png}} | |
| 253 \subfigure[Scratches]{\includegraphics[scale=0.6]{images/Rature_only.png}} | |
| 254 \subfigure[Grey Level \& Contrast]{\includegraphics[scale=0.6]{images/Contrast_only.png}} | |
| 255 \caption{Top left (a): example original image. Others (b-o): examples of the effect | |
| 256 of each transformation module taken separately. Actual perturbed examples are obtained by | |
| 257 a pipeline of these, with random choices about which module to apply and how much perturbation | |
| 258 to apply.} | |
| 259 \label{fig:transform} | |
| 260 %\vspace*{-2mm} | |
| 261 \end{figure*} | |
| 262 | |
| 263 %\vspace*{-3mm} | |
| 264 \section{Experimental Setup} | |
| 265 %\vspace*{-1mm} | |
| 266 | |
| 267 Much previous work on deep learning had been performed on | |
| 268 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009}, | |
| 269 with 60,000 examples, and variants involving 10,000 | |
| 270 examples~\citep{Larochelle-jmlr-2009,VincentPLarochelleH2008-very-small}. | |
| 271 The focus here is on much larger training sets, from 10 times to | |
| 272 to 1000 times larger, and 62 classes. | |
| 273 | |
| 274 The first step in constructing the larger datasets (called NISTP and P07) is to sample from | |
| 275 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas}, | |
| 276 and {\bf OCR data} (scanned machine printed characters). See more in | |
| 277 Section~\ref{sec:sources} below. Once a character | |
| 278 is sampled from one of these sources (chosen randomly), the second step is to | |
| 279 apply a pipeline of transformations and/or noise processes outlined in section \ref{s:perturbations}. | |
| 280 | |
| 281 To provide a baseline of error rate comparison we also estimate human performance | |
| 282 on both the 62-class task and the 10-class digits task. | |
| 283 We compare the best Multi-Layer Perceptrons (MLP) against | |
| 284 the best Stacked Denoising Auto-encoders (SDA), when | |
| 285 both models' hyper-parameters are selected to minimize the validation set error. | |
| 286 We also provide a comparison against a precise estimate | |
| 287 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 288 service ({\tt http://mturk.com}). | |
| 289 AMT users are paid small amounts | |
| 290 of money to perform tasks for which human intelligence is required. | |
| 291 Mechanical Turk has been used extensively in natural language processing and vision. | |
| 292 %processing \citep{SnowEtAl2008} and vision | |
| 293 %\citep{SorokinAndForsyth2008,whitehill09}. | |
| 294 AMT users were presented | |
| 295 with 10 character images (from a test set) on a screen | |
| 296 and asked to label them. | |
| 297 They were forced to choose a single character class (either among the | |
| 298 62 or 10 character classes) for each image. | |
| 299 80 subjects classified 2500 images per (dataset,task) pair. | |
| 300 Different humans labelers sometimes provided a different label for the same | |
| 301 example, and we were able to estimate the error variance due to this effect | |
| 302 because each image was classified by 3 different persons. | |
| 303 The average error of humans on the 62-class task NIST test set | |
| 304 is 18.2\%, with a standard error of 0.1\%. | |
| 305 We controlled noise in the labelling process by (1) | |
| 306 requiring AMT workers with a higher than normal average of accepted | |
| 307 responses ($>$95\%) on other tasks (2) discarding responses that were not | |
| 308 complete (10 predictions) (3) discarding responses for which for which the | |
| 309 time to predict was smaller than 3 seconds for NIST (the mean response time | |
| 310 was 20 seconds) and 6 seconds seconds for NISTP (average response time of | |
| 311 45 seconds) (4) discarding responses which were obviously wrong (10 | |
| 312 identical ones, or "12345..."). Overall, after such filtering, we kept | |
| 313 approximately 95\% of the AMT workers' responses. | |
| 314 | |
| 315 %\vspace*{-3mm} | |
| 316 \subsection{Data Sources} | |
| 317 \label{sec:sources} | |
| 318 %\vspace*{-2mm} | |
| 319 | |
| 320 %\begin{itemize} | |
| 321 %\item | |
| 322 {\bf NIST.} | |
| 323 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995}, | |
| 324 widely used for training and testing character | |
| 325 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}. | |
| 326 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications, | |
| 327 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes | |
| 328 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity. | |
| 329 The fourth partition (called $hsf_4$, 82,587 examples), | |
| 330 experimentally recognized to be the most difficult one, is the one recommended | |
| 331 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} | |
| 332 for that purpose. We randomly split the remainder (731,668 examples) into a training set and a validation set for | |
| 333 model selection. | |
| 334 The performances reported by previous work on that dataset mostly use only the digits. | |
| 335 Here we use all the classes both in the training and testing phase. This is especially | |
| 336 useful to estimate the effect of a multi-task setting. | |
| 337 The distribution of the classes in the NIST training and test sets differs | |
| 338 substantially, with relatively many more digits in the test set, and a more uniform distribution | |
| 339 of letters in the test set (whereas in the training set they are distributed | |
| 340 more like in natural text). | |
| 341 %\vspace*{-1mm} | |
| 342 | |
| 343 %\item | |
| 344 {\bf Fonts.} | |
| 345 In order to have a good variety of sources we downloaded an important number of free fonts from: | |
| 346 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}. | |
| 347 % TODO: pointless to anonymize, it's not pointing to our work | |
| 348 Including an operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from. | |
| 349 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, | |
| 350 directly as input to our models. | |
| 351 %\vspace*{-1mm} | |
| 352 | |
| 353 %\item | |
| 354 {\bf Captchas.} | |
| 355 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a Python-based captcha generator library) for | |
| 356 generating characters of the same format as the NIST dataset. This software is based on | |
| 357 a random character class generator and various kinds of transformations similar to those described in the previous sections. | |
| 358 In order to increase the variability of the data generated, many different fonts are used for generating the characters. | |
| 359 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity | |
| 360 depending on the value of the complexity parameter provided by the user of the data source. | |
| 361 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class? | |
| 362 %\vspace*{-1mm} | |
| 363 | |
| 364 %\item | |
| 365 {\bf OCR data.} | |
| 366 A large set (2 million) of scanned, OCRed and manually verified machine-printed | |
| 367 characters where included as an | |
| 368 additional source. This set is part of a larger corpus being collected by the Image Understanding | |
| 369 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern | |
| 370 ({\tt http://www.iupr.com}), and which will be publicly released. | |
| 371 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this | |
| 372 %\end{itemize} | |
| 373 | |
| 374 %\vspace*{-3mm} | |
| 375 \subsection{Data Sets} | |
| 376 %\vspace*{-2mm} | |
| 377 | |
| 378 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label | |
| 379 from one of the 62 character classes. | |
| 380 %\begin{itemize} | |
| 381 %\vspace*{-1mm} | |
| 382 | |
| 383 %\item | |
| 384 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has | |
| 385 \{651,668 / 80,000 / 82,587\} \{training / validation / test\} examples. | |
| 386 %\vspace*{-1mm} | |
| 387 | |
| 388 %\item | |
| 389 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources | |
| 390 and sending them through the transformation pipeline described in section \ref{s:perturbations}. | |
| 391 For each new example to generate, a data source is selected with probability $10\%$ from the fonts, | |
| 392 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the | |
| 393 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$. | |
| 394 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 395 obtained from the corresponding NIST sets plus other sources. | |
| 396 %\vspace*{-1mm} | |
| 397 | |
| 398 %\item | |
| 399 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources) | |
| 400 except that we only apply | |
| 401 transformations from slant to pinch (see Fig.\ref{fig:transform}(b-f)). | |
| 402 Therefore, the character is | |
| 403 transformed but no additional noise is added to the image, giving images | |
| 404 closer to the NIST dataset. | |
| 405 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 406 obtained from the corresponding NIST sets plus other sources. | |
| 407 %\end{itemize} | |
| 408 | |
| 409 \begin{figure*}[ht] | |
| 410 %\vspace*{-2mm} | |
| 411 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}} | |
| 412 %\vspace*{-2mm} | |
| 413 \caption{Illustration of the computations and training criterion for the denoising | |
| 414 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of | |
| 415 the layer (i.e. raw input or output of previous layer) | |
| 416 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$. | |
| 417 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which | |
| 418 is compared to the uncorrupted input $x$ through the loss function | |
| 419 $L_H(x,z)$, whose expected value is approximately minimized during training | |
| 420 by tuning $\theta$ and $\theta'$.} | |
| 421 \label{fig:da} | |
| 422 %\vspace*{-2mm} | |
| 423 \end{figure*} | |
| 424 | |
| 425 %\vspace*{-3mm} | |
| 426 \subsection{Models and their Hyper-parameters} | |
| 427 %\vspace*{-2mm} | |
| 428 | |
| 429 The experiments are performed using MLPs (with a single | |
| 430 hidden layer) and deep SDAs. | |
| 431 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.} | |
| 432 | |
| 433 {\bf Multi-Layer Perceptrons (MLP).} Whereas previous work had compared | |
| 434 deep architectures to both shallow MLPs and SVMs, we only compared to MLPs | |
| 435 here because of the very large datasets used (making the use of SVMs | |
| 436 computationally challenging because of their quadratic scaling | |
| 437 behavior). Preliminary experiments on training SVMs (libSVM) with subsets | |
| 438 of the training set allowing the program to fit in memory yielded | |
| 439 substantially worse results than those obtained with MLPs\footnote{RBF SVMs | |
| 440 trained with a subset of NISTP or NIST, 100k examples, to fit in memory, | |
| 441 yielded 64\% test error or worse; online linear SVMs trained on the whole | |
| 442 of NIST or 800k from NISTP yielded no better than 42\% error; slightly | |
| 443 better results were obtained by sparsifying the pixel intensities and | |
| 444 projecting to a second-order polynomial (a very sparse vector), still | |
| 445 41\% error. We expect that better results could be obtained with a | |
| 446 better implementation allowing for training with more examples and | |
| 447 a higher-order non-linear projection.} For training on nearly a hundred million examples (with the | |
| 448 perturbed data), the MLPs and SDA are much more convenient than classifiers | |
| 449 based on kernel methods. The MLP has a single hidden layer with $\tanh$ | |
| 450 activation functions, and softmax (normalized exponentials) on the output | |
| 451 layer for estimating $P(class | image)$. The number of hidden units is | |
| 452 taken in $\{300,500,800,1000,1500\}$. Training examples are presented in | |
| 453 minibatches of size 20. A constant learning rate was chosen among $\{0.001, | |
| 454 0.01, 0.025, 0.075, 0.1, 0.5\}$. | |
| 455 %through preliminary experiments (measuring performance on a validation set), | |
| 456 %and $0.1$ (which was found to work best) was then selected for optimizing on | |
| 457 %the whole training sets. | |
| 458 %\vspace*{-1mm} | |
| 459 | |
| 460 | |
| 461 {\bf Stacked Denoising Auto-encoders (SDA).} | |
| 462 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs) | |
| 463 can be used to initialize the weights of each layer of a deep MLP (with many hidden | |
| 464 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006}, | |
| 465 apparently setting parameters in the | |
| 466 basin of attraction of supervised gradient descent yielding better | |
| 467 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised | |
| 468 pre-training phase} uses all of the training images but not the training labels. | |
| 469 Each layer is trained in turn to produce a new representation of its input | |
| 470 (starting from the raw pixels). | |
| 471 It is hypothesized that the | |
| 472 advantage brought by this procedure stems from a better prior, | |
| 473 on the one hand taking advantage of the link between the input | |
| 474 distribution $P(x)$ and the conditional distribution of interest | |
| 475 $P(y|x)$ (like in semi-supervised learning), and on the other hand | |
| 476 taking advantage of the expressive power and bias implicit in the | |
| 477 deep architecture (whereby complex concepts are expressed as | |
| 478 compositions of simpler ones through a deep hierarchy). | |
| 479 | |
| 480 Here we chose to use the Denoising | |
| 481 Auto-encoder~\citep{VincentPLarochelleH2008-very-small} as the building block for | |
| 482 these deep hierarchies of features, as it is simple to train and | |
| 483 explain (see Figure~\ref{fig:da}, as well as | |
| 484 tutorial and code there: {\tt http://deeplearning.net/tutorial}), | |
| 485 provides efficient inference, and yielded results | |
| 486 comparable or better than RBMs in series of experiments | |
| 487 \citep{VincentPLarochelleH2008-very-small}. It really corresponds to a Gaussian | |
| 488 RBM trained by a Score Matching criterion~\cite{Vincent-SM-2010}. | |
| 489 During training, a Denoising | |
| 490 Auto-encoder is presented with a stochastically corrupted version | |
| 491 of the input and trained to reconstruct the uncorrupted input, | |
| 492 forcing the hidden units to represent the leading regularities in | |
| 493 the data. Here we use the random binary masking corruption | |
| 494 (which sets to 0 a random subset of the inputs). | |
| 495 Once it is trained, in a purely unsupervised way, | |
| 496 its hidden units' activations can | |
| 497 be used as inputs for training a second one, etc. | |
| 498 After this unsupervised pre-training stage, the parameters | |
| 499 are used to initialize a deep MLP, which is fine-tuned by | |
| 500 the same standard procedure used to train them (see above). | |
| 501 The SDA hyper-parameters are the same as for the MLP, with the addition of the | |
| 502 amount of corruption noise (we used the masking noise process, whereby a | |
| 503 fixed proportion of the input values, randomly selected, are zeroed), and a | |
| 504 separate learning rate for the unsupervised pre-training stage (selected | |
| 505 from the same above set). The fraction of inputs corrupted was selected | |
| 506 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number | |
| 507 of hidden layers but it was fixed to 3 for most experiments, | |
| 508 based on previous work with | |
| 509 SDAs on MNIST~\citep{VincentPLarochelleH2008-very-small}. | |
| 510 We also compared against 1 and against 2 hidden layers, in order | |
| 511 to disantangle the effect of depth from the effect of unsupervised | |
| 512 pre-training. | |
| 513 The size of the hidden | |
| 514 layers was kept constant across hidden layers, and the best results | |
| 515 were obtained with the largest values that we could experiment | |
| 516 with given our patience, with 1000 hidden units. | |
| 517 | |
| 518 %\vspace*{-1mm} | |
| 519 | |
| 520 \begin{figure*}[ht] | |
| 521 %\vspace*{-2mm} | |
| 522 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}} | |
| 523 %\vspace*{-3mm} | |
| 524 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained | |
| 525 on NIST, 1 on NISTP, and 2 on P07. Left: overall results | |
| 526 of all models, on NIST and NISTP test sets. | |
| 527 Right: error rates on NIST test digits only, along with the previous results from | |
| 528 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005} | |
| 529 respectively based on ART, nearest neighbors, MLPs, and SVMs.} | |
| 530 \label{fig:error-rates-charts} | |
| 531 %\vspace*{-2mm} | |
| 532 \end{figure*} | |
| 533 | |
| 534 | |
| 535 \begin{figure*}[ht] | |
| 536 \vspace*{-3mm} | |
| 537 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}} | |
| 538 \vspace*{-3mm} | |
| 539 \caption{Relative improvement in error rate due to out-of-distribution examples. | |
| 540 Left: Improvement (or loss, when negative) | |
| 541 induced by out-of-distribution examples (perturbed data). | |
| 542 Right: Improvement (or loss, when negative) induced by multi-task | |
| 543 learning (training on all classes and testing only on either digits, | |
| 544 upper case, or lower-case). The deep learner (SDA) benefits more from | |
| 545 out-of-distribution examples, compared to the shallow MLP.} | |
| 546 \label{fig:improvements-charts} | |
| 547 \vspace*{-2mm} | |
| 548 \end{figure*} | |
| 549 | |
| 550 \vspace*{-2mm} | |
| 551 \section{Experimental Results} | |
| 552 \vspace*{-2mm} | |
| 553 | |
| 554 %%\vspace*{-1mm} | |
| 555 %\subsection{SDA vs MLP vs Humans} | |
| 556 %%\vspace*{-1mm} | |
| 557 The models are either trained on NIST (MLP0 and SDA0), | |
| 558 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested | |
| 559 on either NIST, NISTP or P07 (regardless of the data set used for training), | |
| 560 either on the 62-class task | |
| 561 or on the 10-digits task. Training time (including about half | |
| 562 for unsupervised pre-training, for DAs) on the larger | |
| 563 datasets is around one day on a GPU (GTX 285). | |
| 564 Figure~\ref{fig:error-rates-charts} summarizes the results obtained, | |
| 565 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1, | |
| 566 SDA2), along with the previous results on the digits NIST special database | |
| 567 19 test set from the literature, respectively based on ARTMAP neural | |
| 568 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search | |
| 569 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs | |
| 570 ~\citep{Milgram+al-2005}.% More detailed and complete numerical results | |
| 571 %(figures and tables, including standard errors on the error rates) can be | |
| 572 %found in Appendix. | |
| 573 The deep learner not only outperformed the shallow ones and | |
| 574 previously published performance (in a statistically and qualitatively | |
| 575 significant way) but when trained with perturbed data | |
| 576 reaches human performance on both the 62-class task | |
| 577 and the 10-class (digits) task. | |
| 578 17\% error (SDA1) or 18\% error (humans) may seem large but a large | |
| 579 majority of the errors from humans and from SDA1 are from out-of-context | |
| 580 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a | |
| 581 ``c'' and a ``C'' are often indistinguishible). | |
| 582 Regarding shallower networks pre-trained with unsupervised denoising | |
| 583 auto-encders, we find that the NIST test error is 21\% with one hidden | |
| 584 layer and 20\% with two hidden layers (vs 17\% in the same conditions | |
| 585 with 3 hidden layers). Compare this with the 23\% error achieved | |
| 586 by the MLP, i.e. a single hidden layer and no unsupervised pre-training. | |
| 587 As found in previous work~\cite{Erhan+al-2010,Larochelle-jmlr-2009}, | |
| 588 these results show that both depth and | |
| 589 unsupervised pre-training need to be combined in order to achieve | |
| 590 the best results. | |
| 591 | |
| 592 | |
| 593 In addition, as shown in the left of | |
| 594 Figure~\ref{fig:improvements-charts}, the relative improvement in error | |
| 595 rate brought by out-of-distribution examples is greater for the deep | |
| 596 SDA, and these | |
| 597 differences with the shallow MLP are statistically and qualitatively | |
| 598 significant. | |
| 599 The left side of the figure shows the improvement to the clean | |
| 600 NIST test set error brought by the use of out-of-distribution examples | |
| 601 (i.e. the perturbed examples examples from NISTP or P07), | |
| 602 over the models trained exclusively on NIST (respectively SDA0 and MLP0). | |
| 603 Relative percent change is measured by taking | |
| 604 $100 \% \times$ (original model's error / perturbed-data model's error - 1). | |
| 605 The right side of | |
| 606 Figure~\ref{fig:improvements-charts} shows the relative improvement | |
| 607 brought by the use of a multi-task setting, in which the same model is | |
| 608 trained for more classes than the target classes of interest (i.e. training | |
| 609 with all 62 classes when the target classes are respectively the digits, | |
| 610 lower-case, or upper-case characters). Again, whereas the gain from the | |
| 611 multi-task setting is marginal or negative for the MLP, it is substantial | |
| 612 for the SDA. Note that to simplify these multi-task experiments, only the original | |
| 613 NIST dataset is used. For example, the MLP-digits bar shows the relative | |
| 614 percent improvement in MLP error rate on the NIST digits test set | |
| 615 as $100\% \times$ (single-task | |
| 616 model's error / multi-task model's error - 1). The single-task model is | |
| 617 trained with only 10 outputs (one per digit), seeing only digit examples, | |
| 618 whereas the multi-task model is trained with 62 outputs, with all 62 | |
| 619 character classes as examples. Hence the hidden units are shared across | |
| 620 all tasks. For the multi-task model, the digit error rate is measured by | |
| 621 comparing the correct digit class with the output class associated with the | |
| 622 maximum conditional probability among only the digit classes outputs. The | |
| 623 setting is similar for the other two target classes (lower case characters | |
| 624 and upper case characters). Note however that some types of perturbations | |
| 625 (NISTP) help more than others (P07) when testing on the clean images. | |
| 626 %%\vspace*{-1mm} | |
| 627 %\subsection{Perturbed Training Data More Helpful for SDA} | |
| 628 %%\vspace*{-1mm} | |
| 629 | |
| 630 %%\vspace*{-1mm} | |
| 631 %\subsection{Multi-Task Learning Effects} | |
| 632 %%\vspace*{-1mm} | |
| 633 | |
| 634 \iffalse | |
| 635 As previously seen, the SDA is better able to benefit from the | |
| 636 transformations applied to the data than the MLP. In this experiment we | |
| 637 define three tasks: recognizing digits (knowing that the input is a digit), | |
| 638 recognizing upper case characters (knowing that the input is one), and | |
| 639 recognizing lower case characters (knowing that the input is one). We | |
| 640 consider the digit classification task as the target task and we want to | |
| 641 evaluate whether training with the other tasks can help or hurt, and | |
| 642 whether the effect is different for MLPs versus SDAs. The goal is to find | |
| 643 out if deep learning can benefit more (or less) from multiple related tasks | |
| 644 (i.e. the multi-task setting) compared to a corresponding purely supervised | |
| 645 shallow learner. | |
| 646 | |
| 647 We use a single hidden layer MLP with 1000 hidden units, and a SDA | |
| 648 with 3 hidden layers (1000 hidden units per layer), pre-trained and | |
| 649 fine-tuned on NIST. | |
| 650 | |
| 651 Our results show that the MLP benefits marginally from the multi-task setting | |
| 652 in the case of digits (5\% relative improvement) but is actually hurt in the case | |
| 653 of characters (respectively 3\% and 4\% worse for lower and upper class characters). | |
| 654 On the other hand the SDA benefited from the multi-task setting, with relative | |
| 655 error rate improvements of 27\%, 15\% and 13\% respectively for digits, | |
| 656 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. | |
| 657 \fi | |
| 658 | |
| 659 | |
| 660 \vspace*{-2mm} | |
| 661 \section{Conclusions and Discussion} | |
| 662 \vspace*{-2mm} | |
| 663 | |
| 664 We have found that out-of-distribution examples (multi-task learning | |
| 665 and perturbed examples) are more beneficial | |
| 666 to a deep learner than to a traditional shallow and purely | |
| 667 supervised learner. More precisely, | |
| 668 the answers are positive for all the questions asked in the introduction. | |
| 669 %\begin{itemize} | |
| 670 | |
| 671 $\bullet$ %\item | |
| 672 {\bf Do the good results previously obtained with deep architectures on the | |
| 673 MNIST digits generalize to a much larger and richer (but similar) | |
| 674 dataset, the NIST special database 19, with 62 classes and around 800k examples}? | |
| 675 Yes, the SDA {\em systematically outperformed the MLP and all the previously | |
| 676 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level | |
| 677 performance} at around 17\% error on the 62-class task and 1.4\% on the digits, | |
| 678 and beating previously published results on the same data. | |
| 679 | |
| 680 $\bullet$ %\item | |
| 681 {\bf To what extent do out-of-distribution examples help deep learners, | |
| 682 and do they help them more than shallow supervised ones}? | |
| 683 We found that distorted training examples not only made the resulting | |
| 684 classifier better on similarly perturbed images but also on | |
| 685 the {\em original clean examples}, and more importantly and more novel, | |
| 686 that deep architectures benefit more from such {\em out-of-distribution} | |
| 687 examples. Shallow MLPs were helped by perturbed training examples when tested on perturbed input | |
| 688 images (65\% relative improvement on NISTP) | |
| 689 but only marginally helped (5\% relative improvement on all classes) | |
| 690 or even hurt (10\% relative loss on digits) | |
| 691 with respect to clean examples. On the other hand, the deep SDAs | |
| 692 were significantly boosted by these out-of-distribution examples. | |
| 693 Similarly, whereas the improvement due to the multi-task setting was marginal or | |
| 694 negative for the MLP (from +5.6\% to -3.6\% relative change), | |
| 695 it was quite significant for the SDA (from +13\% to +27\% relative change), | |
| 696 which may be explained by the arguments below. | |
| 697 Since out-of-distribution data | |
| 698 (perturbed or from other related classes) is very common, this conclusion | |
| 699 is of practical importance. | |
| 700 %\end{itemize} | |
| 701 | |
| 702 In the original self-taught learning framework~\citep{RainaR2007}, the | |
| 703 out-of-sample examples were used as a source of unsupervised data, and | |
| 704 experiments showed its positive effects in a \emph{limited labeled data} | |
| 705 scenario. However, many of the results by \citet{RainaR2007} (who used a | |
| 706 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught | |
| 707 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases. | |
| 708 We note instead that, for deep | |
| 709 architectures, our experiments show that such a positive effect is accomplished | |
| 710 even in a scenario with a \emph{large number of labeled examples}, | |
| 711 i.e., here, the relative gain of self-taught learning and | |
| 712 out-of-distribution examples is probably preserved | |
| 713 in the asymptotic regime. However, note that in our perturbation experiments | |
| 714 (but not in our multi-task experiments), | |
| 715 even the out-of-distribution examples are labeled, unlike in the | |
| 716 earlier self-taught learning experiments~\citep{RainaR2007}. | |
| 717 | |
| 718 {\bf Why would deep learners benefit more from the self-taught learning | |
| 719 framework and out-of-distribution examples}? | |
| 720 The key idea is that the lower layers of the predictor compute a hierarchy | |
| 721 of features that can be shared across tasks or across variants of the | |
| 722 input distribution. A theoretical analysis of generalization improvements | |
| 723 due to sharing of intermediate features across tasks already points | |
| 724 towards that explanation~\cite{baxter95a}. | |
| 725 Intermediate features that can be used in different | |
| 726 contexts can be estimated in a way that allows to share statistical | |
| 727 strength. Features extracted through many levels are more likely to | |
| 728 be more abstract and more invariant to some of the factors of variation | |
| 729 in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest), | |
| 730 increasing the likelihood that they would be useful for a larger array | |
| 731 of tasks and input conditions. | |
| 732 Therefore, we hypothesize that both depth and unsupervised | |
| 733 pre-training play a part in explaining the advantages observed here, and future | |
| 734 experiments could attempt at teasing apart these factors. | |
| 735 And why would deep learners benefit from the self-taught learning | |
| 736 scenarios even when the number of labeled examples is very large? | |
| 737 We hypothesize that this is related to the hypotheses studied | |
| 738 in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010} | |
| 739 it was found that online learning on a huge dataset did not make the | |
| 740 advantage of the deep learning bias vanish, and a similar phenomenon | |
| 741 may be happening here. We hypothesize that unsupervised pre-training | |
| 742 of a deep hierarchy with out-of-distribution examples initializes the | |
| 743 model in the basin of attraction of supervised gradient descent | |
| 744 that corresponds to better generalization. Furthermore, such good | |
| 745 basins of attraction are not discovered by pure supervised learning | |
| 746 (with or without out-of-distribution examples) from random initialization, and more labeled examples | |
| 747 does not allow the shallow or purely supervised models to discover | |
| 748 the kind of better basins associated | |
| 749 with deep learning and out-of-distribution examples. | |
| 750 | |
| 751 A Flash demo of the recognizer (where both the MLP and the SDA can be compared) | |
| 752 can be executed on-line at the anonymous site {\tt http://deep.host22.com}. | |
| 753 | |
| 754 \iffalse | |
| 755 \section*{Appendix I: Detailed Numerical Results} | |
| 756 | |
| 757 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered. | |
| 758 They also contain additional data such as test errors on P07 and standard errors. | |
| 759 | |
| 760 \begin{table}[ht] | |
| 761 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + | |
| 762 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training | |
| 763 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture | |
| 764 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) | |
| 765 and using a validation set to select hyper-parameters and other training choices. | |
| 766 \{SDA,MLP\}0 are trained on NIST, | |
| 767 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. | |
| 768 The human error rate on digits is a lower bound because it does not count digits that were | |
| 769 recognized as letters. For comparison, the results found in the literature | |
| 770 on NIST digits classification using the same test set are included.} | |
| 771 \label{tab:sda-vs-mlp-vs-humans} | |
| 772 \begin{center} | |
| 773 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 774 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 775 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline | |
| 776 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline | |
| 777 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline | |
| 778 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline | |
| 779 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline | |
| 780 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline | |
| 781 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline | |
| 782 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline | |
| 783 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline | |
| 784 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline | |
| 785 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline | |
| 786 \end{tabular} | |
| 787 \end{center} | |
| 788 \end{table} | |
| 789 | |
| 790 \begin{table}[ht] | |
| 791 \caption{Relative change in error rates due to the use of perturbed training data, | |
| 792 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. | |
| 793 A positive value indicates that training on the perturbed data helped for the | |
| 794 given test set (the first 3 columns on the 62-class tasks and the last one is | |
| 795 on the clean 10-class digits). Clearly, the deep learning models did benefit more | |
| 796 from perturbed training data, even when testing on clean data, whereas the MLP | |
| 797 trained on perturbed data performed worse on the clean digits and about the same | |
| 798 on the clean characters. } | |
| 799 \label{tab:perturbation-effect} | |
| 800 \begin{center} | |
| 801 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 802 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 803 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline | |
| 804 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline | |
| 805 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline | |
| 806 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline | |
| 807 \end{tabular} | |
| 808 \end{center} | |
| 809 \end{table} | |
| 810 | |
| 811 \begin{table}[ht] | |
| 812 \caption{Test error rates and relative change in error rates due to the use of | |
| 813 a multi-task setting, i.e., training on each task in isolation vs training | |
| 814 for all three tasks together, for MLPs vs SDAs. The SDA benefits much | |
| 815 more from the multi-task setting. All experiments on only on the | |
| 816 unperturbed NIST data, using validation error for model selection. | |
| 817 Relative improvement is 1 - single-task error / multi-task error.} | |
| 818 \label{tab:multi-task} | |
| 819 \begin{center} | |
| 820 \begin{tabular}{|l|r|r|r|} \hline | |
| 821 & single-task & multi-task & relative \\ | |
| 822 & setting & setting & improvement \\ \hline | |
| 823 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline | |
| 824 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline | |
| 825 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline | |
| 826 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline | |
| 827 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline | |
| 828 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline | |
| 829 \end{tabular} | |
| 830 \end{center} | |
| 831 \end{table} | |
| 832 | |
| 833 \fi | |
| 834 | |
| 835 %\afterpage{\clearpage} | |
| 836 %\clearpage | |
| 837 { | |
| 838 %\bibliographystyle{spbasic} % basic style, author-year citations | |
| 839 \bibliographystyle{plainnat} | |
| 840 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter} | |
| 841 %\bibliographystyle{unsrtnat} | |
| 842 %\bibliographystyle{apalike} | |
| 843 } | |
| 844 | |
| 845 | |
| 846 \end{document} |