Mercurial > ift6266
annotate writeup/aistats2011_cameraready.tex @ 637:fe98896745a5
fitting
| author | Yoshua Bengio <bengioy@iro.umontreal.ca> |
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| date | Sat, 19 Mar 2011 23:07:03 -0400 |
| parents | 83d53ffe3f25 |
| children | 677d1b1d8158 |
| 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} |
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| 15 \addtolength{\textwidth}{10mm} | |
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| 19 %\setlength\parindent{0mm} | |
| 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 | |
| 634 | 211 algorithms are made available at |
| 212 {\tt http://hg.assembla.com/ift6266}. | |
| 627 | 213 |
| 214 %\vspace*{-3mm} | |
| 215 %\newpage | |
| 216 \section{Perturbed and Transformed Character Images} | |
| 217 \label{s:perturbations} | |
| 218 %\vspace*{-2mm} | |
| 219 | |
| 220 Figure~\ref{fig:transform} shows the different transformations we used to stochastically | |
| 221 transform $32 \times 32$ source images (such as the one in Fig.\ref{fig:torig}) | |
| 222 in order to obtain data from a larger distribution which | |
| 223 covers a domain substantially larger than the clean characters distribution from | |
| 224 which we start. | |
| 225 Although character transformations have been used before to | |
| 226 improve character recognizers, this effort is on a large scale both | |
| 227 in number of classes and in the complexity of the transformations, hence | |
| 228 in the complexity of the learning task. | |
| 229 The code for these transformations (mostly Python) is available at | |
| 634 | 230 {\tt http://hg.assembla.com/ift6266}. All the modules in the pipeline (Figure~\ref{fig:transform}) share |
| 627 | 231 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the |
| 232 amount of deformation or noise introduced. | |
| 233 There are two main parts in the pipeline. The first one, | |
| 234 from thickness to pinch, performs transformations. The second | |
| 235 part, from blur to contrast, adds different kinds of noise. | |
| 634 | 236 More details can be found in~\citep{ARXIV-2010}. |
| 627 | 237 |
| 238 \begin{figure*}[ht] | |
| 239 \centering | |
| 240 \subfigure[Original]{\includegraphics[scale=0.6]{images/Original.png}\label{fig:torig}} | |
| 241 \subfigure[Thickness]{\includegraphics[scale=0.6]{images/Thick_only.png}} | |
| 242 \subfigure[Slant]{\includegraphics[scale=0.6]{images/Slant_only.png}} | |
| 243 \subfigure[Affine Transformation]{\includegraphics[scale=0.6]{images/Affine_only.png}} | |
| 244 \subfigure[Local Elastic Deformation]{\includegraphics[scale=0.6]{images/Localelasticdistorsions_only.png}} | |
| 245 \subfigure[Pinch]{\includegraphics[scale=0.6]{images/Pinch_only.png}} | |
| 246 %Noise | |
| 247 \subfigure[Motion Blur]{\includegraphics[scale=0.6]{images/Motionblur_only.png}} | |
| 248 \subfigure[Occlusion]{\includegraphics[scale=0.6]{images/occlusion_only.png}} | |
| 249 \subfigure[Gaussian Smoothing]{\includegraphics[scale=0.6]{images/Bruitgauss_only.png}} | |
| 250 \subfigure[Pixels Permutation]{\includegraphics[scale=0.6]{images/Permutpixel_only.png}} | |
| 251 \subfigure[Gaussian Noise]{\includegraphics[scale=0.6]{images/Distorsiongauss_only.png}} | |
| 252 \subfigure[Background Image Addition]{\includegraphics[scale=0.6]{images/background_other_only.png}} | |
| 253 \subfigure[Salt \& Pepper]{\includegraphics[scale=0.6]{images/Poivresel_only.png}} | |
| 254 \subfigure[Scratches]{\includegraphics[scale=0.6]{images/Rature_only.png}} | |
| 255 \subfigure[Grey Level \& Contrast]{\includegraphics[scale=0.6]{images/Contrast_only.png}} | |
| 256 \caption{Top left (a): example original image. Others (b-o): examples of the effect | |
| 257 of each transformation module taken separately. Actual perturbed examples are obtained by | |
| 258 a pipeline of these, with random choices about which module to apply and how much perturbation | |
| 259 to apply.} | |
| 260 \label{fig:transform} | |
| 261 %\vspace*{-2mm} | |
| 262 \end{figure*} | |
| 263 | |
| 264 %\vspace*{-3mm} | |
| 265 \section{Experimental Setup} | |
| 266 %\vspace*{-1mm} | |
| 267 | |
| 268 Much previous work on deep learning had been performed on | |
| 269 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009}, | |
| 270 with 60,000 examples, and variants involving 10,000 | |
| 271 examples~\citep{Larochelle-jmlr-2009,VincentPLarochelleH2008-very-small}. | |
| 272 The focus here is on much larger training sets, from 10 times to | |
| 273 to 1000 times larger, and 62 classes. | |
| 274 | |
| 275 The first step in constructing the larger datasets (called NISTP and P07) is to sample from | |
| 276 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas}, | |
| 277 and {\bf OCR data} (scanned machine printed characters). See more in | |
| 278 Section~\ref{sec:sources} below. Once a character | |
| 279 is sampled from one of these sources (chosen randomly), the second step is to | |
| 280 apply a pipeline of transformations and/or noise processes outlined in section \ref{s:perturbations}. | |
| 281 | |
| 282 To provide a baseline of error rate comparison we also estimate human performance | |
| 283 on both the 62-class task and the 10-class digits task. | |
| 284 We compare the best Multi-Layer Perceptrons (MLP) against | |
| 285 the best Stacked Denoising Auto-encoders (SDA), when | |
| 286 both models' hyper-parameters are selected to minimize the validation set error. | |
| 287 We also provide a comparison against a precise estimate | |
| 288 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 289 service ({\tt http://mturk.com}). | |
| 290 AMT users are paid small amounts | |
| 291 of money to perform tasks for which human intelligence is required. | |
| 292 Mechanical Turk has been used extensively in natural language processing and vision. | |
| 293 %processing \citep{SnowEtAl2008} and vision | |
| 294 %\citep{SorokinAndForsyth2008,whitehill09}. | |
| 295 AMT users were presented | |
| 296 with 10 character images (from a test set) on a screen | |
| 297 and asked to label them. | |
| 298 They were forced to choose a single character class (either among the | |
| 299 62 or 10 character classes) for each image. | |
| 300 80 subjects classified 2500 images per (dataset,task) pair. | |
| 301 Different humans labelers sometimes provided a different label for the same | |
| 302 example, and we were able to estimate the error variance due to this effect | |
| 303 because each image was classified by 3 different persons. | |
| 304 The average error of humans on the 62-class task NIST test set | |
| 305 is 18.2\%, with a standard error of 0.1\%. | |
| 306 We controlled noise in the labelling process by (1) | |
| 307 requiring AMT workers with a higher than normal average of accepted | |
| 308 responses ($>$95\%) on other tasks (2) discarding responses that were not | |
| 309 complete (10 predictions) (3) discarding responses for which for which the | |
| 310 time to predict was smaller than 3 seconds for NIST (the mean response time | |
| 311 was 20 seconds) and 6 seconds seconds for NISTP (average response time of | |
| 312 45 seconds) (4) discarding responses which were obviously wrong (10 | |
| 313 identical ones, or "12345..."). Overall, after such filtering, we kept | |
| 314 approximately 95\% of the AMT workers' responses. | |
| 315 | |
| 316 %\vspace*{-3mm} | |
| 317 \subsection{Data Sources} | |
| 318 \label{sec:sources} | |
| 319 %\vspace*{-2mm} | |
| 320 | |
| 321 %\begin{itemize} | |
| 322 %\item | |
| 323 {\bf NIST.} | |
| 324 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995}, | |
| 325 widely used for training and testing character | |
| 326 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}. | |
| 327 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications, | |
| 328 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes | |
| 329 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity. | |
| 330 The fourth partition (called $hsf_4$, 82,587 examples), | |
| 331 experimentally recognized to be the most difficult one, is the one recommended | |
| 332 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} | |
| 333 for that purpose. We randomly split the remainder (731,668 examples) into a training set and a validation set for | |
| 334 model selection. | |
| 335 The performances reported by previous work on that dataset mostly use only the digits. | |
| 336 Here we use all the classes both in the training and testing phase. This is especially | |
| 337 useful to estimate the effect of a multi-task setting. | |
| 338 The distribution of the classes in the NIST training and test sets differs | |
| 339 substantially, with relatively many more digits in the test set, and a more uniform distribution | |
| 340 of letters in the test set (whereas in the training set they are distributed | |
| 341 more like in natural text). | |
| 342 %\vspace*{-1mm} | |
| 343 | |
| 344 %\item | |
| 345 {\bf Fonts.} | |
| 346 In order to have a good variety of sources we downloaded an important number of free fonts from: | |
| 347 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}. | |
| 348 % TODO: pointless to anonymize, it's not pointing to our work | |
| 349 Including an operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from. | |
| 350 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, | |
| 351 directly as input to our models. | |
| 352 %\vspace*{-1mm} | |
| 353 | |
| 354 %\item | |
| 355 {\bf Captchas.} | |
| 356 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a Python-based captcha generator library) for | |
| 357 generating characters of the same format as the NIST dataset. This software is based on | |
| 358 a random character class generator and various kinds of transformations similar to those described in the previous sections. | |
| 359 In order to increase the variability of the data generated, many different fonts are used for generating the characters. | |
| 360 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity | |
| 361 depending on the value of the complexity parameter provided by the user of the data source. | |
| 362 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class? | |
| 363 %\vspace*{-1mm} | |
| 364 | |
| 365 %\item | |
| 366 {\bf OCR data.} | |
| 367 A large set (2 million) of scanned, OCRed and manually verified machine-printed | |
| 368 characters where included as an | |
| 369 additional source. This set is part of a larger corpus being collected by the Image Understanding | |
| 370 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern | |
| 371 ({\tt http://www.iupr.com}), and which will be publicly released. | |
| 372 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this | |
| 373 %\end{itemize} | |
| 374 | |
| 375 %\vspace*{-3mm} | |
| 376 \subsection{Data Sets} | |
| 377 %\vspace*{-2mm} | |
| 378 | |
| 379 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label | |
| 380 from one of the 62 character classes. | |
| 381 %\begin{itemize} | |
| 382 %\vspace*{-1mm} | |
| 383 | |
| 384 %\item | |
| 385 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has | |
| 386 \{651,668 / 80,000 / 82,587\} \{training / validation / test\} examples. | |
| 387 %\vspace*{-1mm} | |
| 388 | |
| 389 %\item | |
| 390 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources | |
| 391 and sending them through the transformation pipeline described in section \ref{s:perturbations}. | |
| 392 For each new example to generate, a data source is selected with probability $10\%$ from the fonts, | |
| 393 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the | |
| 394 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$. | |
| 395 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 396 obtained from the corresponding NIST sets plus other sources. | |
| 397 %\vspace*{-1mm} | |
| 398 | |
| 399 %\item | |
| 400 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources) | |
| 401 except that we only apply | |
| 402 transformations from slant to pinch (see Fig.\ref{fig:transform}(b-f)). | |
| 403 Therefore, the character is | |
| 404 transformed but no additional noise is added to the image, giving images | |
| 405 closer to the NIST dataset. | |
| 406 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 407 obtained from the corresponding NIST sets plus other sources. | |
| 408 %\end{itemize} | |
| 409 | |
| 410 \begin{figure*}[ht] | |
| 411 %\vspace*{-2mm} | |
| 412 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}} | |
| 413 %\vspace*{-2mm} | |
| 414 \caption{Illustration of the computations and training criterion for the denoising | |
| 415 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of | |
| 416 the layer (i.e. raw input or output of previous layer) | |
| 417 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$. | |
| 418 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which | |
| 419 is compared to the uncorrupted input $x$ through the loss function | |
| 420 $L_H(x,z)$, whose expected value is approximately minimized during training | |
| 421 by tuning $\theta$ and $\theta'$.} | |
| 422 \label{fig:da} | |
| 423 %\vspace*{-2mm} | |
| 424 \end{figure*} | |
| 425 | |
| 426 %\vspace*{-3mm} | |
| 427 \subsection{Models and their Hyper-parameters} | |
| 428 %\vspace*{-2mm} | |
| 429 | |
| 430 The experiments are performed using MLPs (with a single | |
| 431 hidden layer) and deep SDAs. | |
| 432 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.} | |
| 433 | |
| 636 | 434 {\bf Multi-Layer Perceptrons (MLP).} The MLP output estimated with |
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435 \[ |
| 636 | 436 P({\rm class}|{\rm input}=x)={\rm softmax}(b_2+W_2\tanh(b_1+W_1 x)), |
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437 \] |
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438 i.e., two layers, where |
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439 \[ |
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440 p={\rm softmax}(a) |
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441 \] |
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442 means that |
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443 \[ |
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444 p_i(x)=\exp(a_i)/\sum_j \exp(a_j) |
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445 \] |
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446 representing the probability |
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447 for class $i$, $\tanh$ is the element-wise |
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448 hyperbolic tangent, $b_i$ are parameter vectors, and $W_i$ are |
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449 parameter matrices (one per layer). The |
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450 number of rows of $W_1$ is called the number of hidden units (of the |
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451 single hidden layer, here), and |
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452 is one way to control capacity (the main other ways to control capacity are |
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453 the number of training iterations and optionally a regularization penalty |
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454 on the parameters, not used here because it did not help). |
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455 Whereas previous work had compared |
| 627 | 456 deep architectures to both shallow MLPs and SVMs, we only compared to MLPs |
| 457 here because of the very large datasets used (making the use of SVMs | |
| 458 computationally challenging because of their quadratic scaling | |
| 459 behavior). Preliminary experiments on training SVMs (libSVM) with subsets | |
| 460 of the training set allowing the program to fit in memory yielded | |
| 461 substantially worse results than those obtained with MLPs\footnote{RBF SVMs | |
| 462 trained with a subset of NISTP or NIST, 100k examples, to fit in memory, | |
| 463 yielded 64\% test error or worse; online linear SVMs trained on the whole | |
| 464 of NIST or 800k from NISTP yielded no better than 42\% error; slightly | |
| 465 better results were obtained by sparsifying the pixel intensities and | |
| 466 projecting to a second-order polynomial (a very sparse vector), still | |
| 467 41\% error. We expect that better results could be obtained with a | |
| 468 better implementation allowing for training with more examples and | |
| 469 a higher-order non-linear projection.} For training on nearly a hundred million examples (with the | |
| 470 perturbed data), the MLPs and SDA are much more convenient than classifiers | |
| 471 based on kernel methods. The MLP has a single hidden layer with $\tanh$ | |
| 472 activation functions, and softmax (normalized exponentials) on the output | |
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473 layer for estimating $P({\rm class} | {\rm input})$. The number of hidden units is |
| 627 | 474 taken in $\{300,500,800,1000,1500\}$. Training examples are presented in |
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475 minibatches of size 20, i.e., the parameters are iteratively updated in the direction |
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476 of the mean gradient of the next 20 examples. A constant learning rate was chosen among $\{0.001, |
| 627 | 477 0.01, 0.025, 0.075, 0.1, 0.5\}$. |
| 478 %through preliminary experiments (measuring performance on a validation set), | |
| 479 %and $0.1$ (which was found to work best) was then selected for optimizing on | |
| 480 %the whole training sets. | |
| 481 %\vspace*{-1mm} | |
| 482 | |
| 483 | |
| 484 {\bf Stacked Denoising Auto-encoders (SDA).} | |
| 485 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs) | |
| 486 can be used to initialize the weights of each layer of a deep MLP (with many hidden | |
| 487 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006}, | |
| 488 apparently setting parameters in the | |
| 489 basin of attraction of supervised gradient descent yielding better | |
| 490 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised | |
| 491 pre-training phase} uses all of the training images but not the training labels. | |
| 492 Each layer is trained in turn to produce a new representation of its input | |
| 493 (starting from the raw pixels). | |
| 494 It is hypothesized that the | |
| 495 advantage brought by this procedure stems from a better prior, | |
| 496 on the one hand taking advantage of the link between the input | |
| 497 distribution $P(x)$ and the conditional distribution of interest | |
| 498 $P(y|x)$ (like in semi-supervised learning), and on the other hand | |
| 499 taking advantage of the expressive power and bias implicit in the | |
| 500 deep architecture (whereby complex concepts are expressed as | |
| 501 compositions of simpler ones through a deep hierarchy). | |
| 502 | |
| 503 Here we chose to use the Denoising | |
| 504 Auto-encoder~\citep{VincentPLarochelleH2008-very-small} as the building block for | |
| 505 these deep hierarchies of features, as it is simple to train and | |
| 506 explain (see Figure~\ref{fig:da}, as well as | |
| 507 tutorial and code there: {\tt http://deeplearning.net/tutorial}), | |
| 508 provides efficient inference, and yielded results | |
| 509 comparable or better than RBMs in series of experiments | |
| 510 \citep{VincentPLarochelleH2008-very-small}. It really corresponds to a Gaussian | |
| 511 RBM trained by a Score Matching criterion~\cite{Vincent-SM-2010}. | |
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512 During its unsupervised training, a Denoising |
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513 Auto-encoder is presented with a stochastically corrupted version $\tilde{x}$ |
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514 of the input $x$ and trained to reconstruct to produce a reconstruction $z$ |
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515 of the uncorrupted input $x$. Because the network has to denoise, it is |
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516 forcing the hidden units $y$ to represent the leading regularities in |
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517 the data. Following~\citep{VincentPLarochelleH2008-very-small} |
| 636 | 518 the hidden units output $y$ is obtained through the sigmoid-affine |
| 519 encoder | |
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520 \[ |
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521 y={\rm sigm}(c+V x) |
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522 \] |
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523 where ${\rm sigm}(a)=1/(1+\exp(-a))$ |
| 636 | 524 and the reconstruction is obtained through the same transformation |
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525 \[ |
| 636 | 526 z={\rm sigm}(d+V' y) |
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527 \] |
| 637 | 528 using the transpose of encoder weights. |
| 529 The training | |
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530 set average of the cross-entropy |
| 637 | 531 reconstruction loss |
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532 \[ |
| 637 | 533 L_H(x,z)=\sum_i z_i \log x_i + (1-z_i) \log(1-x_i) |
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534 \] |
| 637 | 535 is minimized. |
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536 Here we use the random binary masking corruption |
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537 (which in $\tilde{x}$ sets to 0 a random subset of the elements of $x$, and |
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538 copies the rest). |
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539 Once the first denoising auto-encoder is trained, its parameters can be used |
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540 to set the first layer of the deep MLP. The original data are then processed |
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541 through that first layer, and the output of the hidden units form a new |
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542 representation that can be used as input data for training a second denoising |
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543 auto-encoder, still in a purely unsupervised way. |
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544 This is repeated for the desired number of hidden layers. |
| 627 | 545 After this unsupervised pre-training stage, the parameters |
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546 are used to initialize a deep MLP (similar to the above, but |
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547 with more layers), which is fine-tuned by |
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548 the same standard procedure (stochastic gradient descent) |
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549 used to train MLPs in general (see above). |
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550 The top layer parameters of the deep MLP (the one which outputs the |
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551 class probabilities and takes the top hidden layer as input) can |
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552 be initialized at 0. |
| 627 | 553 The SDA hyper-parameters are the same as for the MLP, with the addition of the |
| 554 amount of corruption noise (we used the masking noise process, whereby a | |
| 555 fixed proportion of the input values, randomly selected, are zeroed), and a | |
| 556 separate learning rate for the unsupervised pre-training stage (selected | |
| 557 from the same above set). The fraction of inputs corrupted was selected | |
| 558 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number | |
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559 of hidden layers but it was fixed to 3 for our experiments, |
| 627 | 560 based on previous work with |
| 561 SDAs on MNIST~\citep{VincentPLarochelleH2008-very-small}. | |
| 637 | 562 We also compared against 1 and against 2 hidden layers, |
| 563 to disantangle the effect of depth from that of unsupervised | |
| 627 | 564 pre-training. |
| 637 | 565 The size of each hidden |
| 566 layer was kept constant across hidden layers, and the best results | |
| 567 were obtained with the largest values that we tried | |
| 568 (1000 hidden units). | |
| 627 | 569 |
| 570 %\vspace*{-1mm} | |
| 571 | |
| 572 \begin{figure*}[ht] | |
| 573 %\vspace*{-2mm} | |
| 574 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}} | |
| 575 %\vspace*{-3mm} | |
| 576 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained | |
| 577 on NIST, 1 on NISTP, and 2 on P07. Left: overall results | |
| 578 of all models, on NIST and NISTP test sets. | |
| 579 Right: error rates on NIST test digits only, along with the previous results from | |
| 580 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005} | |
| 581 respectively based on ART, nearest neighbors, MLPs, and SVMs.} | |
| 582 \label{fig:error-rates-charts} | |
| 583 %\vspace*{-2mm} | |
| 584 \end{figure*} | |
| 585 | |
| 586 | |
| 587 \begin{figure*}[ht] | |
| 588 \vspace*{-3mm} | |
| 589 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}} | |
| 590 \vspace*{-3mm} | |
| 591 \caption{Relative improvement in error rate due to out-of-distribution examples. | |
| 592 Left: Improvement (or loss, when negative) | |
| 593 induced by out-of-distribution examples (perturbed data). | |
| 594 Right: Improvement (or loss, when negative) induced by multi-task | |
| 595 learning (training on all classes and testing only on either digits, | |
| 596 upper case, or lower-case). The deep learner (SDA) benefits more from | |
| 597 out-of-distribution examples, compared to the shallow MLP.} | |
| 598 \label{fig:improvements-charts} | |
| 599 \vspace*{-2mm} | |
| 600 \end{figure*} | |
| 601 | |
| 602 \vspace*{-2mm} | |
| 603 \section{Experimental Results} | |
| 604 \vspace*{-2mm} | |
| 605 | |
| 606 %%\vspace*{-1mm} | |
| 607 %\subsection{SDA vs MLP vs Humans} | |
| 608 %%\vspace*{-1mm} | |
| 609 The models are either trained on NIST (MLP0 and SDA0), | |
| 610 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested | |
| 611 on either NIST, NISTP or P07 (regardless of the data set used for training), | |
| 612 either on the 62-class task | |
| 613 or on the 10-digits task. Training time (including about half | |
| 614 for unsupervised pre-training, for DAs) on the larger | |
| 615 datasets is around one day on a GPU (GTX 285). | |
| 616 Figure~\ref{fig:error-rates-charts} summarizes the results obtained, | |
| 617 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1, | |
| 618 SDA2), along with the previous results on the digits NIST special database | |
| 619 19 test set from the literature, respectively based on ARTMAP neural | |
| 620 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search | |
| 621 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs | |
| 622 ~\citep{Milgram+al-2005}.% More detailed and complete numerical results | |
| 623 %(figures and tables, including standard errors on the error rates) can be | |
| 624 %found in Appendix. | |
| 625 The deep learner not only outperformed the shallow ones and | |
| 626 previously published performance (in a statistically and qualitatively | |
| 627 significant way) but when trained with perturbed data | |
| 628 reaches human performance on both the 62-class task | |
| 629 and the 10-class (digits) task. | |
| 630 17\% error (SDA1) or 18\% error (humans) may seem large but a large | |
| 631 majority of the errors from humans and from SDA1 are from out-of-context | |
| 632 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a | |
| 633 ``c'' and a ``C'' are often indistinguishible). | |
| 634 Regarding shallower networks pre-trained with unsupervised denoising | |
| 635 auto-encders, we find that the NIST test error is 21\% with one hidden | |
| 636 layer and 20\% with two hidden layers (vs 17\% in the same conditions | |
| 637 with 3 hidden layers). Compare this with the 23\% error achieved | |
| 638 by the MLP, i.e. a single hidden layer and no unsupervised pre-training. | |
| 639 As found in previous work~\cite{Erhan+al-2010,Larochelle-jmlr-2009}, | |
| 640 these results show that both depth and | |
| 641 unsupervised pre-training need to be combined in order to achieve | |
| 642 the best results. | |
| 643 | |
| 644 | |
| 645 In addition, as shown in the left of | |
| 646 Figure~\ref{fig:improvements-charts}, the relative improvement in error | |
| 647 rate brought by out-of-distribution examples is greater for the deep | |
| 648 SDA, and these | |
| 649 differences with the shallow MLP are statistically and qualitatively | |
| 650 significant. | |
| 651 The left side of the figure shows the improvement to the clean | |
| 652 NIST test set error brought by the use of out-of-distribution examples | |
| 653 (i.e. the perturbed examples examples from NISTP or P07), | |
| 654 over the models trained exclusively on NIST (respectively SDA0 and MLP0). | |
| 655 Relative percent change is measured by taking | |
| 656 $100 \% \times$ (original model's error / perturbed-data model's error - 1). | |
| 657 The right side of | |
| 658 Figure~\ref{fig:improvements-charts} shows the relative improvement | |
| 659 brought by the use of a multi-task setting, in which the same model is | |
| 660 trained for more classes than the target classes of interest (i.e. training | |
| 661 with all 62 classes when the target classes are respectively the digits, | |
| 662 lower-case, or upper-case characters). Again, whereas the gain from the | |
| 663 multi-task setting is marginal or negative for the MLP, it is substantial | |
| 664 for the SDA. Note that to simplify these multi-task experiments, only the original | |
| 665 NIST dataset is used. For example, the MLP-digits bar shows the relative | |
| 666 percent improvement in MLP error rate on the NIST digits test set | |
| 667 as $100\% \times$ (single-task | |
| 668 model's error / multi-task model's error - 1). The single-task model is | |
| 669 trained with only 10 outputs (one per digit), seeing only digit examples, | |
| 670 whereas the multi-task model is trained with 62 outputs, with all 62 | |
| 671 character classes as examples. Hence the hidden units are shared across | |
| 672 all tasks. For the multi-task model, the digit error rate is measured by | |
| 673 comparing the correct digit class with the output class associated with the | |
| 674 maximum conditional probability among only the digit classes outputs. The | |
| 675 setting is similar for the other two target classes (lower case characters | |
| 676 and upper case characters). Note however that some types of perturbations | |
| 677 (NISTP) help more than others (P07) when testing on the clean images. | |
| 678 %%\vspace*{-1mm} | |
| 679 %\subsection{Perturbed Training Data More Helpful for SDA} | |
| 680 %%\vspace*{-1mm} | |
| 681 | |
| 682 %%\vspace*{-1mm} | |
| 683 %\subsection{Multi-Task Learning Effects} | |
| 684 %%\vspace*{-1mm} | |
| 685 | |
| 686 \iffalse | |
| 687 As previously seen, the SDA is better able to benefit from the | |
| 688 transformations applied to the data than the MLP. In this experiment we | |
| 689 define three tasks: recognizing digits (knowing that the input is a digit), | |
| 690 recognizing upper case characters (knowing that the input is one), and | |
| 691 recognizing lower case characters (knowing that the input is one). We | |
| 692 consider the digit classification task as the target task and we want to | |
| 693 evaluate whether training with the other tasks can help or hurt, and | |
| 694 whether the effect is different for MLPs versus SDAs. The goal is to find | |
| 695 out if deep learning can benefit more (or less) from multiple related tasks | |
| 696 (i.e. the multi-task setting) compared to a corresponding purely supervised | |
| 697 shallow learner. | |
| 698 | |
| 699 We use a single hidden layer MLP with 1000 hidden units, and a SDA | |
| 700 with 3 hidden layers (1000 hidden units per layer), pre-trained and | |
| 701 fine-tuned on NIST. | |
| 702 | |
| 703 Our results show that the MLP benefits marginally from the multi-task setting | |
| 704 in the case of digits (5\% relative improvement) but is actually hurt in the case | |
| 705 of characters (respectively 3\% and 4\% worse for lower and upper class characters). | |
| 706 On the other hand the SDA benefited from the multi-task setting, with relative | |
| 707 error rate improvements of 27\%, 15\% and 13\% respectively for digits, | |
| 708 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. | |
| 709 \fi | |
| 710 | |
| 711 | |
| 712 \vspace*{-2mm} | |
| 713 \section{Conclusions and Discussion} | |
| 714 \vspace*{-2mm} | |
| 715 | |
| 716 We have found that out-of-distribution examples (multi-task learning | |
| 717 and perturbed examples) are more beneficial | |
| 718 to a deep learner than to a traditional shallow and purely | |
| 719 supervised learner. More precisely, | |
| 720 the answers are positive for all the questions asked in the introduction. | |
| 721 %\begin{itemize} | |
| 722 | |
| 723 $\bullet$ %\item | |
| 724 {\bf Do the good results previously obtained with deep architectures on the | |
| 725 MNIST digits generalize to a much larger and richer (but similar) | |
| 726 dataset, the NIST special database 19, with 62 classes and around 800k examples}? | |
| 727 Yes, the SDA {\em systematically outperformed the MLP and all the previously | |
| 728 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level | |
| 729 performance} at around 17\% error on the 62-class task and 1.4\% on the digits, | |
| 730 and beating previously published results on the same data. | |
| 731 | |
| 732 $\bullet$ %\item | |
| 733 {\bf To what extent do out-of-distribution examples help deep learners, | |
| 734 and do they help them more than shallow supervised ones}? | |
| 735 We found that distorted training examples not only made the resulting | |
| 736 classifier better on similarly perturbed images but also on | |
| 737 the {\em original clean examples}, and more importantly and more novel, | |
| 738 that deep architectures benefit more from such {\em out-of-distribution} | |
| 739 examples. Shallow MLPs were helped by perturbed training examples when tested on perturbed input | |
| 740 images (65\% relative improvement on NISTP) | |
| 741 but only marginally helped (5\% relative improvement on all classes) | |
| 742 or even hurt (10\% relative loss on digits) | |
| 743 with respect to clean examples. On the other hand, the deep SDAs | |
| 744 were significantly boosted by these out-of-distribution examples. | |
| 745 Similarly, whereas the improvement due to the multi-task setting was marginal or | |
| 746 negative for the MLP (from +5.6\% to -3.6\% relative change), | |
| 747 it was quite significant for the SDA (from +13\% to +27\% relative change), | |
| 748 which may be explained by the arguments below. | |
| 749 Since out-of-distribution data | |
| 750 (perturbed or from other related classes) is very common, this conclusion | |
| 751 is of practical importance. | |
| 752 %\end{itemize} | |
| 753 | |
| 754 In the original self-taught learning framework~\citep{RainaR2007}, the | |
| 755 out-of-sample examples were used as a source of unsupervised data, and | |
| 756 experiments showed its positive effects in a \emph{limited labeled data} | |
| 757 scenario. However, many of the results by \citet{RainaR2007} (who used a | |
| 758 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught | |
| 759 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases. | |
| 760 We note instead that, for deep | |
| 761 architectures, our experiments show that such a positive effect is accomplished | |
| 762 even in a scenario with a \emph{large number of labeled examples}, | |
| 763 i.e., here, the relative gain of self-taught learning and | |
| 764 out-of-distribution examples is probably preserved | |
| 765 in the asymptotic regime. However, note that in our perturbation experiments | |
| 766 (but not in our multi-task experiments), | |
| 767 even the out-of-distribution examples are labeled, unlike in the | |
| 768 earlier self-taught learning experiments~\citep{RainaR2007}. | |
| 769 | |
| 770 {\bf Why would deep learners benefit more from the self-taught learning | |
| 771 framework and out-of-distribution examples}? | |
| 772 The key idea is that the lower layers of the predictor compute a hierarchy | |
| 773 of features that can be shared across tasks or across variants of the | |
| 774 input distribution. A theoretical analysis of generalization improvements | |
| 775 due to sharing of intermediate features across tasks already points | |
| 776 towards that explanation~\cite{baxter95a}. | |
| 777 Intermediate features that can be used in different | |
| 778 contexts can be estimated in a way that allows to share statistical | |
| 779 strength. Features extracted through many levels are more likely to | |
| 780 be more abstract and more invariant to some of the factors of variation | |
| 781 in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest), | |
| 782 increasing the likelihood that they would be useful for a larger array | |
| 783 of tasks and input conditions. | |
| 784 Therefore, we hypothesize that both depth and unsupervised | |
| 785 pre-training play a part in explaining the advantages observed here, and future | |
| 786 experiments could attempt at teasing apart these factors. | |
| 787 And why would deep learners benefit from the self-taught learning | |
| 788 scenarios even when the number of labeled examples is very large? | |
| 789 We hypothesize that this is related to the hypotheses studied | |
| 790 in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010} | |
| 791 it was found that online learning on a huge dataset did not make the | |
| 792 advantage of the deep learning bias vanish, and a similar phenomenon | |
| 793 may be happening here. We hypothesize that unsupervised pre-training | |
| 794 of a deep hierarchy with out-of-distribution examples initializes the | |
| 795 model in the basin of attraction of supervised gradient descent | |
| 796 that corresponds to better generalization. Furthermore, such good | |
| 797 basins of attraction are not discovered by pure supervised learning | |
| 798 (with or without out-of-distribution examples) from random initialization, and more labeled examples | |
| 799 does not allow the shallow or purely supervised models to discover | |
| 800 the kind of better basins associated | |
| 801 with deep learning and out-of-distribution examples. | |
| 802 | |
| 803 A Flash demo of the recognizer (where both the MLP and the SDA can be compared) | |
| 634 | 804 can be executed on-line at {\tt http://deep.host22.com}. |
| 627 | 805 |
| 806 \iffalse | |
| 807 \section*{Appendix I: Detailed Numerical Results} | |
| 808 | |
| 809 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered. | |
| 810 They also contain additional data such as test errors on P07 and standard errors. | |
| 811 | |
| 812 \begin{table}[ht] | |
| 813 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + | |
| 814 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training | |
| 815 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture | |
| 816 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) | |
| 817 and using a validation set to select hyper-parameters and other training choices. | |
| 818 \{SDA,MLP\}0 are trained on NIST, | |
| 819 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. | |
| 820 The human error rate on digits is a lower bound because it does not count digits that were | |
| 821 recognized as letters. For comparison, the results found in the literature | |
| 822 on NIST digits classification using the same test set are included.} | |
| 823 \label{tab:sda-vs-mlp-vs-humans} | |
| 824 \begin{center} | |
| 825 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 826 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 827 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline | |
| 828 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline | |
| 829 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline | |
| 830 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline | |
| 831 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline | |
| 832 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline | |
| 833 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline | |
| 834 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline | |
| 835 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline | |
| 836 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline | |
| 837 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline | |
| 838 \end{tabular} | |
| 839 \end{center} | |
| 840 \end{table} | |
| 841 | |
| 842 \begin{table}[ht] | |
| 843 \caption{Relative change in error rates due to the use of perturbed training data, | |
| 844 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. | |
| 845 A positive value indicates that training on the perturbed data helped for the | |
| 846 given test set (the first 3 columns on the 62-class tasks and the last one is | |
| 847 on the clean 10-class digits). Clearly, the deep learning models did benefit more | |
| 848 from perturbed training data, even when testing on clean data, whereas the MLP | |
| 849 trained on perturbed data performed worse on the clean digits and about the same | |
| 850 on the clean characters. } | |
| 851 \label{tab:perturbation-effect} | |
| 852 \begin{center} | |
| 853 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 854 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 855 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline | |
| 856 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline | |
| 857 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline | |
| 858 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline | |
| 859 \end{tabular} | |
| 860 \end{center} | |
| 861 \end{table} | |
| 862 | |
| 863 \begin{table}[ht] | |
| 864 \caption{Test error rates and relative change in error rates due to the use of | |
| 865 a multi-task setting, i.e., training on each task in isolation vs training | |
| 866 for all three tasks together, for MLPs vs SDAs. The SDA benefits much | |
| 867 more from the multi-task setting. All experiments on only on the | |
| 868 unperturbed NIST data, using validation error for model selection. | |
| 869 Relative improvement is 1 - single-task error / multi-task error.} | |
| 870 \label{tab:multi-task} | |
| 871 \begin{center} | |
| 872 \begin{tabular}{|l|r|r|r|} \hline | |
| 873 & single-task & multi-task & relative \\ | |
| 874 & setting & setting & improvement \\ \hline | |
| 875 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline | |
| 876 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline | |
| 877 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline | |
| 878 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline | |
| 879 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline | |
| 880 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline | |
| 881 \end{tabular} | |
| 882 \end{center} | |
| 883 \end{table} | |
| 884 | |
| 885 \fi | |
| 886 | |
| 887 %\afterpage{\clearpage} | |
| 888 %\clearpage | |
| 889 { | |
| 890 %\bibliographystyle{spbasic} % basic style, author-year citations | |
| 891 \bibliographystyle{plainnat} | |
| 892 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter} | |
| 893 %\bibliographystyle{unsrtnat} | |
| 894 %\bibliographystyle{apalike} | |
| 895 } | |
| 896 | |
| 897 | |
| 898 \end{document} |