Mercurial > ift6266
comparison writeup/aistats2011_cameraready.tex @ 627:249a180795e3
camera ready version
| author | Yoshua Bengio <bengioy@iro.umontreal.ca> |
|---|---|
| date | Thu, 17 Mar 2011 08:15:34 -0400 |
| parents | |
| children | f55f1b1499c4 |
comparison
equal
deleted
inserted
replaced
| 624:49933073590c | 627:249a180795e3 |
|---|---|
| 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} | |
| 13 \usepackage[numbers]{natbib} | |
| 14 | |
| 15 \addtolength{\textwidth}{10mm} | |
| 16 \addtolength{\evensidemargin}{-5mm} | |
| 17 \addtolength{\oddsidemargin}{-5mm} | |
| 18 | |
| 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 | |
| 30 Arnaud Bergeron \and | |
| 31 Nicolas Boulanger-Lewandowski \and | |
| 32 Thomas Breuel \and | |
| 33 Youssouf Chherawala \and | |
| 34 Moustapha Cisse \and | |
| 35 Myriam Côté \and | |
| 36 Dumitru Erhan \and | |
| 37 Jeremy Eustache \and | |
| 38 Xavier Glorot \and | |
| 39 Xavier Muller \and | |
| 40 Sylvain Pannetier Lebeuf \and | |
| 41 Razvan Pascanu \and | |
| 42 Salah Rifai \and | |
| 43 Francois Savard \and | |
| 44 Guillaume Sicard | |
| 45 \vspace*{5mm}}] | |
| 46 \aistatsaddress{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} | |
| 47 %\date{{\tt bengioy@iro.umontreal.ca}, Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada} | |
| 48 %\jmlrheading{}{2010}{}{10/2010}{XX/2011}{Yoshua Bengio et al} | |
| 49 %\editor{} | |
| 50 | |
| 51 %\makeanontitle | |
| 52 %\maketitle | |
| 53 | |
| 54 %{\bf Running title: Deep Self-Taught Learning} | |
| 55 | |
| 56 \vspace*{5mm} | |
| 57 \begin{abstract} | |
| 58 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}. | |
| 59 \end{abstract} | |
| 60 %\vspace*{-3mm} | |
| 61 | |
| 62 %\begin{keywords} | |
| 63 %Deep learning, self-taught learning, out-of-distribution examples, handwritten character recognition, multi-task learning | |
| 64 %\end{keywords} | |
| 65 %\keywords{self-taught learning \and multi-task learning \and out-of-distribution examples \and deep learning \and handwriting recognition} | |
| 66 | |
| 67 | |
| 68 | |
| 69 \section{Introduction} | |
| 70 %\vspace*{-1mm} | |
| 71 | |
| 72 {\bf Deep Learning} has emerged as a promising new area of research in | |
| 73 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. | |
| 74 Learning algorithms for deep architectures are centered on the learning | |
| 75 of useful representations of data, which are better suited to the task at hand, | |
| 76 and are organized in a hierarchy with multiple levels. | |
| 77 This is in part inspired by observations of the mammalian visual cortex, | |
| 78 which consists of a chain of processing elements, each of which is associated with a | |
| 79 different representation of the raw visual input. In fact, | |
| 80 it was found recently that the features learnt in deep architectures resemble | |
| 81 those observed in the first two of these stages (in areas V1 and V2 | |
| 82 of visual cortex) \citep{HonglakL2008}, and that they become more and | |
| 83 more invariant to factors of variation (such as camera movement) in | |
| 84 higher layers~\citep{Goodfellow2009}. | |
| 85 It has been hypothesized that learning a hierarchy of features increases the | |
| 86 ease and practicality of developing representations that are at once | |
| 87 tailored to specific tasks, yet are able to borrow statistical strength | |
| 88 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the | |
| 89 feature representation can lead to higher-level (more abstract, more | |
| 90 general) features that are more robust to unanticipated sources of | |
| 91 variance extant in real data. | |
| 92 | |
| 93 Whereas a deep architecture can in principle be more powerful than a | |
| 94 shallow one in terms of representation, depth appears to render the | |
| 95 training problem more difficult in terms of optimization and local minima. | |
| 96 It is also only recently that successful algorithms were proposed to | |
| 97 overcome some of these difficulties. All are based on unsupervised | |
| 98 learning, often in an greedy layer-wise ``unsupervised pre-training'' | |
| 99 stage~\citep{Bengio-2009}. | |
| 100 The principle is that each layer starting from | |
| 101 the bottom is trained to represent its input (the output of the previous | |
| 102 layer). After this | |
| 103 unsupervised initialization, the stack of layers can be | |
| 104 converted into a deep supervised feedforward neural network and fine-tuned by | |
| 105 stochastic gradient descent. | |
| 106 One of these layer initialization techniques, | |
| 107 applied here, is the Denoising | |
| 108 Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see | |
| 109 Figure~\ref{fig:da}), which performed similarly or | |
| 110 better~\citep{VincentPLarochelleH2008-very-small} than previously | |
| 111 proposed Restricted Boltzmann Machines (RBM)~\citep{Hinton06} | |
| 112 in terms of unsupervised extraction | |
| 113 of a hierarchy of features useful for classification. Each layer is trained | |
| 114 to denoise its input, creating a layer of features that can be used as | |
| 115 input for the next layer, forming a Stacked Denoising Auto-encoder (SDA). | |
| 116 Note that training a Denoising Auto-encoder | |
| 117 can actually been seen as training a particular RBM by an inductive | |
| 118 principle different from maximum likelihood~\citep{Vincent-SM-2010}, | |
| 119 namely by Score Matching~\citep{Hyvarinen-2005,HyvarinenA2008}. | |
| 120 | |
| 121 Previous comparative experimental results with stacking of RBMs and DAs | |
| 122 to build deep supervised predictors had shown that they could outperform | |
| 123 shallow architectures in a variety of settings, especially | |
| 124 when the data involves complex interactions between many factors of | |
| 125 variation~\citep{LarochelleH2007,Bengio-2009}. Other experiments have suggested | |
| 126 that the unsupervised layer-wise pre-training acted as a useful | |
| 127 prior~\citep{Erhan+al-2010} that allows one to initialize a deep | |
| 128 neural network in a relatively much smaller region of parameter space, | |
| 129 corresponding to better generalization. | |
| 130 | |
| 131 To further the understanding of the reasons for the good performance | |
| 132 observed with deep learners, we focus here on the following {\em hypothesis}: | |
| 133 intermediate levels of representation, especially when there are | |
| 134 more such levels, can be exploited to {\bf share | |
| 135 statistical strength across different but related types of examples}, | |
| 136 such as examples coming from other tasks than the task of interest | |
| 137 (the multi-task setting), or examples coming from an overlapping | |
| 138 but different distribution (images with different kinds of perturbations | |
| 139 and noises, here). This is consistent with the hypotheses discussed | |
| 140 in~\citet{Bengio-2009} regarding the potential advantage | |
| 141 of deep learning and the idea that more levels of representation can | |
| 142 give rise to more abstract, more general features of the raw input. | |
| 143 | |
| 144 This hypothesis is related to a learning setting called | |
| 145 {\bf self-taught learning}~\citep{RainaR2007}, which combines principles | |
| 146 of semi-supervised and multi-task learning: the learner can exploit examples | |
| 147 that are unlabeled and possibly come from a distribution different from the target | |
| 148 distribution, e.g., from other classes than those of interest. | |
| 149 It has already been shown that deep learners can clearly take advantage of | |
| 150 unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small}, | |
| 151 but more needed to be done to explore the impact | |
| 152 of {\em out-of-distribution} examples and of the {\em multi-task} setting | |
| 153 (one exception is~\citep{CollobertR2008}, which shares and uses unsupervised | |
| 154 pre-training only with the first layer). In particular the {\em relative | |
| 155 advantage of deep learning} for these settings has not been evaluated. | |
| 156 | |
| 157 | |
| 158 % | |
| 159 The {\bf main claim} of this paper is that deep learners (with several levels of representation) can | |
| 160 {\bf benefit more from out-of-distribution examples than shallow learners} (with a single | |
| 161 level), both in the context of the multi-task setting and from | |
| 162 perturbed examples. Because we are able to improve on state-of-the-art | |
| 163 performance and reach human-level performance | |
| 164 on a large-scale task, we consider that this paper is also a contribution | |
| 165 to advance the application of machine learning to handwritten character recognition. | |
| 166 More precisely, we ask and answer the following questions: | |
| 167 | |
| 168 %\begin{enumerate} | |
| 169 $\bullet$ %\item | |
| 170 Do the good results previously obtained with deep architectures on the | |
| 171 MNIST digit images generalize to the setting of a similar but much larger and richer | |
| 172 dataset, the NIST special database 19, with 62 classes and around 800k examples? | |
| 173 | |
| 174 $\bullet$ %\item | |
| 175 To what extent does the perturbation of input images (e.g. adding | |
| 176 noise, affine transformations, background images) make the resulting | |
| 177 classifiers better not only on similarly perturbed images but also on | |
| 178 the {\em original clean examples}? We study this question in the | |
| 179 context of the 62-class and 10-class tasks of the NIST special database 19. | |
| 180 | |
| 181 $\bullet$ %\item | |
| 182 Do deep architectures {\em benefit {\bf more} from such out-of-distribution} | |
| 183 examples, in particular do they benefit more from | |
| 184 examples that are perturbed versions of the examples from the task of interest? | |
| 185 | |
| 186 $\bullet$ %\item | |
| 187 Similarly, does the feature learning step in deep learning algorithms benefit {\bf more} | |
| 188 from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than | |
| 189 a corresponding shallow and purely supervised architecture? | |
| 190 We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case) | |
| 191 to answer this question. | |
| 192 %\end{enumerate} | |
| 193 | |
| 194 Our experimental results provide positive evidence towards all of these questions, | |
| 195 as well as {\bf classifiers that reach human-level performance on 62-class isolated character | |
| 196 recognition and beat previously published results on the NIST dataset (special database 19)}. | |
| 197 To achieve these results, we introduce in the next section a sophisticated system | |
| 198 for stochastically transforming character images and then explain the methodology, | |
| 199 which is based on training with or without these transformed images and testing on | |
| 200 clean ones. | |
| 201 Code for generating these transformations as well as for the deep learning | |
| 202 algorithms are made available at {\tt http://anonymous.url.net}.%{\tt http://hg.assembla.com/ift6266}. | |
| 203 | |
| 204 %\vspace*{-3mm} | |
| 205 %\newpage | |
| 206 \section{Perturbed and Transformed Character Images} | |
| 207 \label{s:perturbations} | |
| 208 %\vspace*{-2mm} | |
| 209 | |
| 210 Figure~\ref{fig:transform} shows the different transformations we used to stochastically | |
| 211 transform $32 \times 32$ source images (such as the one in Fig.\ref{fig:torig}) | |
| 212 in order to obtain data from a larger distribution which | |
| 213 covers a domain substantially larger than the clean characters distribution from | |
| 214 which we start. | |
| 215 Although character transformations have been used before to | |
| 216 improve character recognizers, this effort is on a large scale both | |
| 217 in number of classes and in the complexity of the transformations, hence | |
| 218 in the complexity of the learning task. | |
| 219 The code for these transformations (mostly Python) is available at | |
| 220 {\tt http://anonymous.url.net}. All the modules in the pipeline (Figure~\ref{fig:transform}) share | |
| 221 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the | |
| 222 amount of deformation or noise introduced. | |
| 223 There are two main parts in the pipeline. The first one, | |
| 224 from thickness to pinch, performs transformations. The second | |
| 225 part, from blur to contrast, adds different kinds of noise. | |
| 226 More details can be found in~\citep{ift6266-tr-anonymous}. | |
| 227 | |
| 228 \begin{figure*}[ht] | |
| 229 \centering | |
| 230 \subfigure[Original]{\includegraphics[scale=0.6]{images/Original.png}\label{fig:torig}} | |
| 231 \subfigure[Thickness]{\includegraphics[scale=0.6]{images/Thick_only.png}} | |
| 232 \subfigure[Slant]{\includegraphics[scale=0.6]{images/Slant_only.png}} | |
| 233 \subfigure[Affine Transformation]{\includegraphics[scale=0.6]{images/Affine_only.png}} | |
| 234 \subfigure[Local Elastic Deformation]{\includegraphics[scale=0.6]{images/Localelasticdistorsions_only.png}} | |
| 235 \subfigure[Pinch]{\includegraphics[scale=0.6]{images/Pinch_only.png}} | |
| 236 %Noise | |
| 237 \subfigure[Motion Blur]{\includegraphics[scale=0.6]{images/Motionblur_only.png}} | |
| 238 \subfigure[Occlusion]{\includegraphics[scale=0.6]{images/occlusion_only.png}} | |
| 239 \subfigure[Gaussian Smoothing]{\includegraphics[scale=0.6]{images/Bruitgauss_only.png}} | |
| 240 \subfigure[Pixels Permutation]{\includegraphics[scale=0.6]{images/Permutpixel_only.png}} | |
| 241 \subfigure[Gaussian Noise]{\includegraphics[scale=0.6]{images/Distorsiongauss_only.png}} | |
| 242 \subfigure[Background Image Addition]{\includegraphics[scale=0.6]{images/background_other_only.png}} | |
| 243 \subfigure[Salt \& Pepper]{\includegraphics[scale=0.6]{images/Poivresel_only.png}} | |
| 244 \subfigure[Scratches]{\includegraphics[scale=0.6]{images/Rature_only.png}} | |
| 245 \subfigure[Grey Level \& Contrast]{\includegraphics[scale=0.6]{images/Contrast_only.png}} | |
| 246 \caption{Top left (a): example original image. Others (b-o): examples of the effect | |
| 247 of each transformation module taken separately. Actual perturbed examples are obtained by | |
| 248 a pipeline of these, with random choices about which module to apply and how much perturbation | |
| 249 to apply.} | |
| 250 \label{fig:transform} | |
| 251 %\vspace*{-2mm} | |
| 252 \end{figure*} | |
| 253 | |
| 254 %\vspace*{-3mm} | |
| 255 \section{Experimental Setup} | |
| 256 %\vspace*{-1mm} | |
| 257 | |
| 258 Much previous work on deep learning had been performed on | |
| 259 the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009}, | |
| 260 with 60,000 examples, and variants involving 10,000 | |
| 261 examples~\citep{Larochelle-jmlr-2009,VincentPLarochelleH2008-very-small}. | |
| 262 The focus here is on much larger training sets, from 10 times to | |
| 263 to 1000 times larger, and 62 classes. | |
| 264 | |
| 265 The first step in constructing the larger datasets (called NISTP and P07) is to sample from | |
| 266 a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas}, | |
| 267 and {\bf OCR data} (scanned machine printed characters). See more in | |
| 268 Section~\ref{sec:sources} below. Once a character | |
| 269 is sampled from one of these sources (chosen randomly), the second step is to | |
| 270 apply a pipeline of transformations and/or noise processes outlined in section \ref{s:perturbations}. | |
| 271 | |
| 272 To provide a baseline of error rate comparison we also estimate human performance | |
| 273 on both the 62-class task and the 10-class digits task. | |
| 274 We compare the best Multi-Layer Perceptrons (MLP) against | |
| 275 the best Stacked Denoising Auto-encoders (SDA), when | |
| 276 both models' hyper-parameters are selected to minimize the validation set error. | |
| 277 We also provide a comparison against a precise estimate | |
| 278 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 279 service ({\tt http://mturk.com}). | |
| 280 AMT users are paid small amounts | |
| 281 of money to perform tasks for which human intelligence is required. | |
| 282 Mechanical Turk has been used extensively in natural language processing and vision. | |
| 283 %processing \citep{SnowEtAl2008} and vision | |
| 284 %\citep{SorokinAndForsyth2008,whitehill09}. | |
| 285 AMT users were presented | |
| 286 with 10 character images (from a test set) on a screen | |
| 287 and asked to label them. | |
| 288 They were forced to choose a single character class (either among the | |
| 289 62 or 10 character classes) for each image. | |
| 290 80 subjects classified 2500 images per (dataset,task) pair. | |
| 291 Different humans labelers sometimes provided a different label for the same | |
| 292 example, and we were able to estimate the error variance due to this effect | |
| 293 because each image was classified by 3 different persons. | |
| 294 The average error of humans on the 62-class task NIST test set | |
| 295 is 18.2\%, with a standard error of 0.1\%. | |
| 296 We controlled noise in the labelling process by (1) | |
| 297 requiring AMT workers with a higher than normal average of accepted | |
| 298 responses ($>$95\%) on other tasks (2) discarding responses that were not | |
| 299 complete (10 predictions) (3) discarding responses for which for which the | |
| 300 time to predict was smaller than 3 seconds for NIST (the mean response time | |
| 301 was 20 seconds) and 6 seconds seconds for NISTP (average response time of | |
| 302 45 seconds) (4) discarding responses which were obviously wrong (10 | |
| 303 identical ones, or "12345..."). Overall, after such filtering, we kept | |
| 304 approximately 95\% of the AMT workers' responses. | |
| 305 | |
| 306 %\vspace*{-3mm} | |
| 307 \subsection{Data Sources} | |
| 308 \label{sec:sources} | |
| 309 %\vspace*{-2mm} | |
| 310 | |
| 311 %\begin{itemize} | |
| 312 %\item | |
| 313 {\bf NIST.} | |
| 314 Our main source of characters is the NIST Special Database 19~\citep{Grother-1995}, | |
| 315 widely used for training and testing character | |
| 316 recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}. | |
| 317 The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications, | |
| 318 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes | |
| 319 corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity. | |
| 320 The fourth partition (called $hsf_4$, 82,587 examples), | |
| 321 experimentally recognized to be the most difficult one, is the one recommended | |
| 322 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} | |
| 323 for that purpose. We randomly split the remainder (731,668 examples) into a training set and a validation set for | |
| 324 model selection. | |
| 325 The performances reported by previous work on that dataset mostly use only the digits. | |
| 326 Here we use all the classes both in the training and testing phase. This is especially | |
| 327 useful to estimate the effect of a multi-task setting. | |
| 328 The distribution of the classes in the NIST training and test sets differs | |
| 329 substantially, with relatively many more digits in the test set, and a more uniform distribution | |
| 330 of letters in the test set (whereas in the training set they are distributed | |
| 331 more like in natural text). | |
| 332 %\vspace*{-1mm} | |
| 333 | |
| 334 %\item | |
| 335 {\bf Fonts.} | |
| 336 In order to have a good variety of sources we downloaded an important number of free fonts from: | |
| 337 {\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}. | |
| 338 % TODO: pointless to anonymize, it's not pointing to our work | |
| 339 Including an operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from. | |
| 340 The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, | |
| 341 directly as input to our models. | |
| 342 %\vspace*{-1mm} | |
| 343 | |
| 344 %\item | |
| 345 {\bf Captchas.} | |
| 346 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a Python-based captcha generator library) for | |
| 347 generating characters of the same format as the NIST dataset. This software is based on | |
| 348 a random character class generator and various kinds of transformations similar to those described in the previous sections. | |
| 349 In order to increase the variability of the data generated, many different fonts are used for generating the characters. | |
| 350 Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity | |
| 351 depending on the value of the complexity parameter provided by the user of the data source. | |
| 352 %Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class? | |
| 353 %\vspace*{-1mm} | |
| 354 | |
| 355 %\item | |
| 356 {\bf OCR data.} | |
| 357 A large set (2 million) of scanned, OCRed and manually verified machine-printed | |
| 358 characters where included as an | |
| 359 additional source. This set is part of a larger corpus being collected by the Image Understanding | |
| 360 Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern | |
| 361 ({\tt http://www.iupr.com}), and which will be publicly released. | |
| 362 %TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this | |
| 363 %\end{itemize} | |
| 364 | |
| 365 %\vspace*{-3mm} | |
| 366 \subsection{Data Sets} | |
| 367 %\vspace*{-2mm} | |
| 368 | |
| 369 All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label | |
| 370 from one of the 62 character classes. | |
| 371 %\begin{itemize} | |
| 372 %\vspace*{-1mm} | |
| 373 | |
| 374 %\item | |
| 375 {\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has | |
| 376 \{651,668 / 80,000 / 82,587\} \{training / validation / test\} examples. | |
| 377 %\vspace*{-1mm} | |
| 378 | |
| 379 %\item | |
| 380 {\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources | |
| 381 and sending them through the transformation pipeline described in section \ref{s:perturbations}. | |
| 382 For each new example to generate, a data source is selected with probability $10\%$ from the fonts, | |
| 383 $25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the | |
| 384 order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$. | |
| 385 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 386 obtained from the corresponding NIST sets plus other sources. | |
| 387 %\vspace*{-1mm} | |
| 388 | |
| 389 %\item | |
| 390 {\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources) | |
| 391 except that we only apply | |
| 392 transformations from slant to pinch (see Fig.\ref{fig:transform}(b-f)). | |
| 393 Therefore, the character is | |
| 394 transformed but no additional noise is added to the image, giving images | |
| 395 closer to the NIST dataset. | |
| 396 It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples | |
| 397 obtained from the corresponding NIST sets plus other sources. | |
| 398 %\end{itemize} | |
| 399 | |
| 400 \begin{figure*}[ht] | |
| 401 %\vspace*{-2mm} | |
| 402 \centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}} | |
| 403 %\vspace*{-2mm} | |
| 404 \caption{Illustration of the computations and training criterion for the denoising | |
| 405 auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of | |
| 406 the layer (i.e. raw input or output of previous layer) | |
| 407 s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$. | |
| 408 The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which | |
| 409 is compared to the uncorrupted input $x$ through the loss function | |
| 410 $L_H(x,z)$, whose expected value is approximately minimized during training | |
| 411 by tuning $\theta$ and $\theta'$.} | |
| 412 \label{fig:da} | |
| 413 %\vspace*{-2mm} | |
| 414 \end{figure*} | |
| 415 | |
| 416 %\vspace*{-3mm} | |
| 417 \subsection{Models and their Hyper-parameters} | |
| 418 %\vspace*{-2mm} | |
| 419 | |
| 420 The experiments are performed using MLPs (with a single | |
| 421 hidden layer) and deep SDAs. | |
| 422 \emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.} | |
| 423 | |
| 424 {\bf Multi-Layer Perceptrons (MLP).} Whereas previous work had compared | |
| 425 deep architectures to both shallow MLPs and SVMs, we only compared to MLPs | |
| 426 here because of the very large datasets used (making the use of SVMs | |
| 427 computationally challenging because of their quadratic scaling | |
| 428 behavior). Preliminary experiments on training SVMs (libSVM) with subsets | |
| 429 of the training set allowing the program to fit in memory yielded | |
| 430 substantially worse results than those obtained with MLPs\footnote{RBF SVMs | |
| 431 trained with a subset of NISTP or NIST, 100k examples, to fit in memory, | |
| 432 yielded 64\% test error or worse; online linear SVMs trained on the whole | |
| 433 of NIST or 800k from NISTP yielded no better than 42\% error; slightly | |
| 434 better results were obtained by sparsifying the pixel intensities and | |
| 435 projecting to a second-order polynomial (a very sparse vector), still | |
| 436 41\% error. We expect that better results could be obtained with a | |
| 437 better implementation allowing for training with more examples and | |
| 438 a higher-order non-linear projection.} For training on nearly a hundred million examples (with the | |
| 439 perturbed data), the MLPs and SDA are much more convenient than classifiers | |
| 440 based on kernel methods. The MLP has a single hidden layer with $\tanh$ | |
| 441 activation functions, and softmax (normalized exponentials) on the output | |
| 442 layer for estimating $P(class | image)$. The number of hidden units is | |
| 443 taken in $\{300,500,800,1000,1500\}$. Training examples are presented in | |
| 444 minibatches of size 20. A constant learning rate was chosen among $\{0.001, | |
| 445 0.01, 0.025, 0.075, 0.1, 0.5\}$. | |
| 446 %through preliminary experiments (measuring performance on a validation set), | |
| 447 %and $0.1$ (which was found to work best) was then selected for optimizing on | |
| 448 %the whole training sets. | |
| 449 %\vspace*{-1mm} | |
| 450 | |
| 451 | |
| 452 {\bf Stacked Denoising Auto-encoders (SDA).} | |
| 453 Various auto-encoder variants and Restricted Boltzmann Machines (RBMs) | |
| 454 can be used to initialize the weights of each layer of a deep MLP (with many hidden | |
| 455 layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006}, | |
| 456 apparently setting parameters in the | |
| 457 basin of attraction of supervised gradient descent yielding better | |
| 458 generalization~\citep{Erhan+al-2010}. This initial {\em unsupervised | |
| 459 pre-training phase} uses all of the training images but not the training labels. | |
| 460 Each layer is trained in turn to produce a new representation of its input | |
| 461 (starting from the raw pixels). | |
| 462 It is hypothesized that the | |
| 463 advantage brought by this procedure stems from a better prior, | |
| 464 on the one hand taking advantage of the link between the input | |
| 465 distribution $P(x)$ and the conditional distribution of interest | |
| 466 $P(y|x)$ (like in semi-supervised learning), and on the other hand | |
| 467 taking advantage of the expressive power and bias implicit in the | |
| 468 deep architecture (whereby complex concepts are expressed as | |
| 469 compositions of simpler ones through a deep hierarchy). | |
| 470 | |
| 471 Here we chose to use the Denoising | |
| 472 Auto-encoder~\citep{VincentPLarochelleH2008-very-small} as the building block for | |
| 473 these deep hierarchies of features, as it is simple to train and | |
| 474 explain (see Figure~\ref{fig:da}, as well as | |
| 475 tutorial and code there: {\tt http://deeplearning.net/tutorial}), | |
| 476 provides efficient inference, and yielded results | |
| 477 comparable or better than RBMs in series of experiments | |
| 478 \citep{VincentPLarochelleH2008-very-small}. It really corresponds to a Gaussian | |
| 479 RBM trained by a Score Matching criterion~\cite{Vincent-SM-2010}. | |
| 480 During training, a Denoising | |
| 481 Auto-encoder is presented with a stochastically corrupted version | |
| 482 of the input and trained to reconstruct the uncorrupted input, | |
| 483 forcing the hidden units to represent the leading regularities in | |
| 484 the data. Here we use the random binary masking corruption | |
| 485 (which sets to 0 a random subset of the inputs). | |
| 486 Once it is trained, in a purely unsupervised way, | |
| 487 its hidden units' activations can | |
| 488 be used as inputs for training a second one, etc. | |
| 489 After this unsupervised pre-training stage, the parameters | |
| 490 are used to initialize a deep MLP, which is fine-tuned by | |
| 491 the same standard procedure used to train them (see above). | |
| 492 The SDA hyper-parameters are the same as for the MLP, with the addition of the | |
| 493 amount of corruption noise (we used the masking noise process, whereby a | |
| 494 fixed proportion of the input values, randomly selected, are zeroed), and a | |
| 495 separate learning rate for the unsupervised pre-training stage (selected | |
| 496 from the same above set). The fraction of inputs corrupted was selected | |
| 497 among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number | |
| 498 of hidden layers but it was fixed to 3 for most experiments, | |
| 499 based on previous work with | |
| 500 SDAs on MNIST~\citep{VincentPLarochelleH2008-very-small}. | |
| 501 We also compared against 1 and against 2 hidden layers, in order | |
| 502 to disantangle the effect of depth from the effect of unsupervised | |
| 503 pre-training. | |
| 504 The size of the hidden | |
| 505 layers was kept constant across hidden layers, and the best results | |
| 506 were obtained with the largest values that we could experiment | |
| 507 with given our patience, with 1000 hidden units. | |
| 508 | |
| 509 %\vspace*{-1mm} | |
| 510 | |
| 511 \begin{figure*}[ht] | |
| 512 %\vspace*{-2mm} | |
| 513 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}} | |
| 514 %\vspace*{-3mm} | |
| 515 \caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained | |
| 516 on NIST, 1 on NISTP, and 2 on P07. Left: overall results | |
| 517 of all models, on NIST and NISTP test sets. | |
| 518 Right: error rates on NIST test digits only, along with the previous results from | |
| 519 literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005} | |
| 520 respectively based on ART, nearest neighbors, MLPs, and SVMs.} | |
| 521 \label{fig:error-rates-charts} | |
| 522 %\vspace*{-2mm} | |
| 523 \end{figure*} | |
| 524 | |
| 525 | |
| 526 \begin{figure*}[ht] | |
| 527 \vspace*{-3mm} | |
| 528 \centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}} | |
| 529 \vspace*{-3mm} | |
| 530 \caption{Relative improvement in error rate due to out-of-distribution examples. | |
| 531 Left: Improvement (or loss, when negative) | |
| 532 induced by out-of-distribution examples (perturbed data). | |
| 533 Right: Improvement (or loss, when negative) induced by multi-task | |
| 534 learning (training on all classes and testing only on either digits, | |
| 535 upper case, or lower-case). The deep learner (SDA) benefits more from | |
| 536 out-of-distribution examples, compared to the shallow MLP.} | |
| 537 \label{fig:improvements-charts} | |
| 538 \vspace*{-2mm} | |
| 539 \end{figure*} | |
| 540 | |
| 541 \vspace*{-2mm} | |
| 542 \section{Experimental Results} | |
| 543 \vspace*{-2mm} | |
| 544 | |
| 545 %%\vspace*{-1mm} | |
| 546 %\subsection{SDA vs MLP vs Humans} | |
| 547 %%\vspace*{-1mm} | |
| 548 The models are either trained on NIST (MLP0 and SDA0), | |
| 549 NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested | |
| 550 on either NIST, NISTP or P07 (regardless of the data set used for training), | |
| 551 either on the 62-class task | |
| 552 or on the 10-digits task. Training time (including about half | |
| 553 for unsupervised pre-training, for DAs) on the larger | |
| 554 datasets is around one day on a GPU (GTX 285). | |
| 555 Figure~\ref{fig:error-rates-charts} summarizes the results obtained, | |
| 556 comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1, | |
| 557 SDA2), along with the previous results on the digits NIST special database | |
| 558 19 test set from the literature, respectively based on ARTMAP neural | |
| 559 networks ~\citep{Granger+al-2007}, fast nearest-neighbor search | |
| 560 ~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs | |
| 561 ~\citep{Milgram+al-2005}.% More detailed and complete numerical results | |
| 562 %(figures and tables, including standard errors on the error rates) can be | |
| 563 %found in Appendix. | |
| 564 The deep learner not only outperformed the shallow ones and | |
| 565 previously published performance (in a statistically and qualitatively | |
| 566 significant way) but when trained with perturbed data | |
| 567 reaches human performance on both the 62-class task | |
| 568 and the 10-class (digits) task. | |
| 569 17\% error (SDA1) or 18\% error (humans) may seem large but a large | |
| 570 majority of the errors from humans and from SDA1 are from out-of-context | |
| 571 confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a | |
| 572 ``c'' and a ``C'' are often indistinguishible). | |
| 573 Regarding shallower networks pre-trained with unsupervised denoising | |
| 574 auto-encders, we find that the NIST test error is 21\% with one hidden | |
| 575 layer and 20\% with two hidden layers (vs 17\% in the same conditions | |
| 576 with 3 hidden layers). Compare this with the 23\% error achieved | |
| 577 by the MLP, i.e. a single hidden layer and no unsupervised pre-training. | |
| 578 As found in previous work~\cite{Erhan+al-2010,Larochelle-jmlr-2009}, | |
| 579 these results show that both depth and | |
| 580 unsupervised pre-training need to be combined in order to achieve | |
| 581 the best results. | |
| 582 | |
| 583 | |
| 584 In addition, as shown in the left of | |
| 585 Figure~\ref{fig:improvements-charts}, the relative improvement in error | |
| 586 rate brought by out-of-distribution examples is greater for the deep | |
| 587 SDA, and these | |
| 588 differences with the shallow MLP are statistically and qualitatively | |
| 589 significant. | |
| 590 The left side of the figure shows the improvement to the clean | |
| 591 NIST test set error brought by the use of out-of-distribution examples | |
| 592 (i.e. the perturbed examples examples from NISTP or P07), | |
| 593 over the models trained exclusively on NIST (respectively SDA0 and MLP0). | |
| 594 Relative percent change is measured by taking | |
| 595 $100 \% \times$ (original model's error / perturbed-data model's error - 1). | |
| 596 The right side of | |
| 597 Figure~\ref{fig:improvements-charts} shows the relative improvement | |
| 598 brought by the use of a multi-task setting, in which the same model is | |
| 599 trained for more classes than the target classes of interest (i.e. training | |
| 600 with all 62 classes when the target classes are respectively the digits, | |
| 601 lower-case, or upper-case characters). Again, whereas the gain from the | |
| 602 multi-task setting is marginal or negative for the MLP, it is substantial | |
| 603 for the SDA. Note that to simplify these multi-task experiments, only the original | |
| 604 NIST dataset is used. For example, the MLP-digits bar shows the relative | |
| 605 percent improvement in MLP error rate on the NIST digits test set | |
| 606 as $100\% \times$ (single-task | |
| 607 model's error / multi-task model's error - 1). The single-task model is | |
| 608 trained with only 10 outputs (one per digit), seeing only digit examples, | |
| 609 whereas the multi-task model is trained with 62 outputs, with all 62 | |
| 610 character classes as examples. Hence the hidden units are shared across | |
| 611 all tasks. For the multi-task model, the digit error rate is measured by | |
| 612 comparing the correct digit class with the output class associated with the | |
| 613 maximum conditional probability among only the digit classes outputs. The | |
| 614 setting is similar for the other two target classes (lower case characters | |
| 615 and upper case characters). Note however that some types of perturbations | |
| 616 (NISTP) help more than others (P07) when testing on the clean images. | |
| 617 %%\vspace*{-1mm} | |
| 618 %\subsection{Perturbed Training Data More Helpful for SDA} | |
| 619 %%\vspace*{-1mm} | |
| 620 | |
| 621 %%\vspace*{-1mm} | |
| 622 %\subsection{Multi-Task Learning Effects} | |
| 623 %%\vspace*{-1mm} | |
| 624 | |
| 625 \iffalse | |
| 626 As previously seen, the SDA is better able to benefit from the | |
| 627 transformations applied to the data than the MLP. In this experiment we | |
| 628 define three tasks: recognizing digits (knowing that the input is a digit), | |
| 629 recognizing upper case characters (knowing that the input is one), and | |
| 630 recognizing lower case characters (knowing that the input is one). We | |
| 631 consider the digit classification task as the target task and we want to | |
| 632 evaluate whether training with the other tasks can help or hurt, and | |
| 633 whether the effect is different for MLPs versus SDAs. The goal is to find | |
| 634 out if deep learning can benefit more (or less) from multiple related tasks | |
| 635 (i.e. the multi-task setting) compared to a corresponding purely supervised | |
| 636 shallow learner. | |
| 637 | |
| 638 We use a single hidden layer MLP with 1000 hidden units, and a SDA | |
| 639 with 3 hidden layers (1000 hidden units per layer), pre-trained and | |
| 640 fine-tuned on NIST. | |
| 641 | |
| 642 Our results show that the MLP benefits marginally from the multi-task setting | |
| 643 in the case of digits (5\% relative improvement) but is actually hurt in the case | |
| 644 of characters (respectively 3\% and 4\% worse for lower and upper class characters). | |
| 645 On the other hand the SDA benefited from the multi-task setting, with relative | |
| 646 error rate improvements of 27\%, 15\% and 13\% respectively for digits, | |
| 647 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. | |
| 648 \fi | |
| 649 | |
| 650 | |
| 651 \vspace*{-2mm} | |
| 652 \section{Conclusions and Discussion} | |
| 653 \vspace*{-2mm} | |
| 654 | |
| 655 We have found that out-of-distribution examples (multi-task learning | |
| 656 and perturbed examples) are more beneficial | |
| 657 to a deep learner than to a traditional shallow and purely | |
| 658 supervised learner. More precisely, | |
| 659 the answers are positive for all the questions asked in the introduction. | |
| 660 %\begin{itemize} | |
| 661 | |
| 662 $\bullet$ %\item | |
| 663 {\bf Do the good results previously obtained with deep architectures on the | |
| 664 MNIST digits generalize to a much larger and richer (but similar) | |
| 665 dataset, the NIST special database 19, with 62 classes and around 800k examples}? | |
| 666 Yes, the SDA {\em systematically outperformed the MLP and all the previously | |
| 667 published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level | |
| 668 performance} at around 17\% error on the 62-class task and 1.4\% on the digits, | |
| 669 and beating previously published results on the same data. | |
| 670 | |
| 671 $\bullet$ %\item | |
| 672 {\bf To what extent do out-of-distribution examples help deep learners, | |
| 673 and do they help them more than shallow supervised ones}? | |
| 674 We found that distorted training examples not only made the resulting | |
| 675 classifier better on similarly perturbed images but also on | |
| 676 the {\em original clean examples}, and more importantly and more novel, | |
| 677 that deep architectures benefit more from such {\em out-of-distribution} | |
| 678 examples. Shallow MLPs were helped by perturbed training examples when tested on perturbed input | |
| 679 images (65\% relative improvement on NISTP) | |
| 680 but only marginally helped (5\% relative improvement on all classes) | |
| 681 or even hurt (10\% relative loss on digits) | |
| 682 with respect to clean examples. On the other hand, the deep SDAs | |
| 683 were significantly boosted by these out-of-distribution examples. | |
| 684 Similarly, whereas the improvement due to the multi-task setting was marginal or | |
| 685 negative for the MLP (from +5.6\% to -3.6\% relative change), | |
| 686 it was quite significant for the SDA (from +13\% to +27\% relative change), | |
| 687 which may be explained by the arguments below. | |
| 688 Since out-of-distribution data | |
| 689 (perturbed or from other related classes) is very common, this conclusion | |
| 690 is of practical importance. | |
| 691 %\end{itemize} | |
| 692 | |
| 693 In the original self-taught learning framework~\citep{RainaR2007}, the | |
| 694 out-of-sample examples were used as a source of unsupervised data, and | |
| 695 experiments showed its positive effects in a \emph{limited labeled data} | |
| 696 scenario. However, many of the results by \citet{RainaR2007} (who used a | |
| 697 shallow, sparse coding approach) suggest that the {\em relative gain of self-taught | |
| 698 learning vs ordinary supervised learning} diminishes as the number of labeled examples increases. | |
| 699 We note instead that, for deep | |
| 700 architectures, our experiments show that such a positive effect is accomplished | |
| 701 even in a scenario with a \emph{large number of labeled examples}, | |
| 702 i.e., here, the relative gain of self-taught learning and | |
| 703 out-of-distribution examples is probably preserved | |
| 704 in the asymptotic regime. However, note that in our perturbation experiments | |
| 705 (but not in our multi-task experiments), | |
| 706 even the out-of-distribution examples are labeled, unlike in the | |
| 707 earlier self-taught learning experiments~\citep{RainaR2007}. | |
| 708 | |
| 709 {\bf Why would deep learners benefit more from the self-taught learning | |
| 710 framework and out-of-distribution examples}? | |
| 711 The key idea is that the lower layers of the predictor compute a hierarchy | |
| 712 of features that can be shared across tasks or across variants of the | |
| 713 input distribution. A theoretical analysis of generalization improvements | |
| 714 due to sharing of intermediate features across tasks already points | |
| 715 towards that explanation~\cite{baxter95a}. | |
| 716 Intermediate features that can be used in different | |
| 717 contexts can be estimated in a way that allows to share statistical | |
| 718 strength. Features extracted through many levels are more likely to | |
| 719 be more abstract and more invariant to some of the factors of variation | |
| 720 in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest), | |
| 721 increasing the likelihood that they would be useful for a larger array | |
| 722 of tasks and input conditions. | |
| 723 Therefore, we hypothesize that both depth and unsupervised | |
| 724 pre-training play a part in explaining the advantages observed here, and future | |
| 725 experiments could attempt at teasing apart these factors. | |
| 726 And why would deep learners benefit from the self-taught learning | |
| 727 scenarios even when the number of labeled examples is very large? | |
| 728 We hypothesize that this is related to the hypotheses studied | |
| 729 in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010} | |
| 730 it was found that online learning on a huge dataset did not make the | |
| 731 advantage of the deep learning bias vanish, and a similar phenomenon | |
| 732 may be happening here. We hypothesize that unsupervised pre-training | |
| 733 of a deep hierarchy with out-of-distribution examples initializes the | |
| 734 model in the basin of attraction of supervised gradient descent | |
| 735 that corresponds to better generalization. Furthermore, such good | |
| 736 basins of attraction are not discovered by pure supervised learning | |
| 737 (with or without out-of-distribution examples) from random initialization, and more labeled examples | |
| 738 does not allow the shallow or purely supervised models to discover | |
| 739 the kind of better basins associated | |
| 740 with deep learning and out-of-distribution examples. | |
| 741 | |
| 742 A Flash demo of the recognizer (where both the MLP and the SDA can be compared) | |
| 743 can be executed on-line at the anonymous site {\tt http://deep.host22.com}. | |
| 744 | |
| 745 \iffalse | |
| 746 \section*{Appendix I: Detailed Numerical Results} | |
| 747 | |
| 748 These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered. | |
| 749 They also contain additional data such as test errors on P07 and standard errors. | |
| 750 | |
| 751 \begin{table}[ht] | |
| 752 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + | |
| 753 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training | |
| 754 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture | |
| 755 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) | |
| 756 and using a validation set to select hyper-parameters and other training choices. | |
| 757 \{SDA,MLP\}0 are trained on NIST, | |
| 758 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. | |
| 759 The human error rate on digits is a lower bound because it does not count digits that were | |
| 760 recognized as letters. For comparison, the results found in the literature | |
| 761 on NIST digits classification using the same test set are included.} | |
| 762 \label{tab:sda-vs-mlp-vs-humans} | |
| 763 \begin{center} | |
| 764 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 765 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 766 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $1.4\%$ \\ \hline | |
| 767 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline | |
| 768 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline | |
| 769 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline | |
| 770 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline | |
| 771 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline | |
| 772 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline | |
| 773 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline | |
| 774 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline | |
| 775 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline | |
| 776 \citep{Milgram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline | |
| 777 \end{tabular} | |
| 778 \end{center} | |
| 779 \end{table} | |
| 780 | |
| 781 \begin{table}[ht] | |
| 782 \caption{Relative change in error rates due to the use of perturbed training data, | |
| 783 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. | |
| 784 A positive value indicates that training on the perturbed data helped for the | |
| 785 given test set (the first 3 columns on the 62-class tasks and the last one is | |
| 786 on the clean 10-class digits). Clearly, the deep learning models did benefit more | |
| 787 from perturbed training data, even when testing on clean data, whereas the MLP | |
| 788 trained on perturbed data performed worse on the clean digits and about the same | |
| 789 on the clean characters. } | |
| 790 \label{tab:perturbation-effect} | |
| 791 \begin{center} | |
| 792 \begin{tabular}{|l|r|r|r|r|} \hline | |
| 793 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline | |
| 794 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline | |
| 795 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline | |
| 796 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline | |
| 797 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline | |
| 798 \end{tabular} | |
| 799 \end{center} | |
| 800 \end{table} | |
| 801 | |
| 802 \begin{table}[ht] | |
| 803 \caption{Test error rates and relative change in error rates due to the use of | |
| 804 a multi-task setting, i.e., training on each task in isolation vs training | |
| 805 for all three tasks together, for MLPs vs SDAs. The SDA benefits much | |
| 806 more from the multi-task setting. All experiments on only on the | |
| 807 unperturbed NIST data, using validation error for model selection. | |
| 808 Relative improvement is 1 - single-task error / multi-task error.} | |
| 809 \label{tab:multi-task} | |
| 810 \begin{center} | |
| 811 \begin{tabular}{|l|r|r|r|} \hline | |
| 812 & single-task & multi-task & relative \\ | |
| 813 & setting & setting & improvement \\ \hline | |
| 814 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline | |
| 815 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline | |
| 816 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline | |
| 817 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline | |
| 818 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline | |
| 819 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline | |
| 820 \end{tabular} | |
| 821 \end{center} | |
| 822 \end{table} | |
| 823 | |
| 824 \fi | |
| 825 | |
| 826 %\afterpage{\clearpage} | |
| 827 %\clearpage | |
| 828 { | |
| 829 %\bibliographystyle{spbasic} % basic style, author-year citations | |
| 830 \bibliographystyle{plainnat} | |
| 831 \bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter} | |
| 832 %\bibliographystyle{unsrtnat} | |
| 833 %\bibliographystyle{apalike} | |
| 834 } | |
| 835 | |
| 836 | |
| 837 \end{document} |