ift6266: writeup/aistats2011_revised.tex comparison

comparison writeup/aistats2011_revised.tex @ 633:13baba8a4522

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author	Yoshua Bengio <bengioy@iro.umontreal.ca>
date	Sat, 19 Mar 2011 22:51:40 -0400
parents	49933073590c
children

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+%\documentclass[twoside,11pt]{article} % For LaTeX2e
+\documentclass{article} % For LaTeX2e
+\usepackage{aistats2e_2011}
+%\usepackage{times}
+\usepackage{wrapfig}
+\usepackage{amsthm}
+\usepackage{amsmath}
+\usepackage{bbm}
+\usepackage[utf8]{inputenc}
+\usepackage[psamsfonts]{amssymb}
+%\usepackage{algorithm,algorithmic} % not used after all
+\usepackage{graphicx,subfigure}
+\usepackage[numbers]{natbib}
+\addtolength{\textwidth}{10mm}
+\addtolength{\evensidemargin}{-5mm}
+\addtolength{\oddsidemargin}{-5mm}
+%\setlength\parindent{0mm}
+\begin{document}
+\twocolumn[
+\aistatstitle{Deep Learners Benefit More from Out-of-Distribution Examples}
+\runningtitle{Deep Learners for Out-of-Distribution Examples}
+\runningauthor{Bengio et. al.}
+\aistatsauthor{Anonymous Authors\\
+\vspace*{5mm}}]
+\iffalse
+Yoshua  Bengio \and
+Frédéric  Bastien \and
+Arnaud  Bergeron \and
+Nicolas  Boulanger-Lewandowski \and
+Thomas  Breuel \and
+Youssouf  Chherawala \and
+Moustapha  Cisse \and
+Myriam  Côté \and
+Dumitru  Erhan \and
+Jeremy  Eustache \and
+Xavier  Glorot \and
+Xavier  Muller \and
+Sylvain  Pannetier Lebeuf \and
+Razvan  Pascanu \and
+Salah  Rifai \and
+Francois  Savard \and
+Guillaume  Sicard
+%}
+\fi
+%\aistatsaddress{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada}
+%\date{{\tt bengioy@iro.umontreal.ca}, Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada}
+%\jmlrheading{}{2010}{}{10/2010}{XX/2011}{Yoshua Bengio et al}
+%\editor{}
+%\makeanontitle
+%\maketitle
+%{\bf Running title: Deep Self-Taught Learning}
+\vspace*{5mm}
+\begin{abstract}
+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}.
+\end{abstract}
+%\vspace*{-3mm}
+%\begin{keywords}
+%Deep learning, self-taught learning, out-of-distribution examples, handwritten character recognition, multi-task learning
+%\end{keywords}
+%\keywords{self-taught learning \and multi-task learning \and out-of-distribution examples \and deep learning \and handwriting recognition}
+\section{Introduction}
+%\vspace*{-1mm}
+{\bf Deep Learning} has emerged as a promising new area of research in
+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.
+Learning algorithms for deep architectures are centered on the learning
+of useful representations of data, which are better suited to the task at hand,
+and are organized in a hierarchy with multiple levels.
+This is in part inspired by observations of the mammalian visual cortex,
+which consists of a chain of processing elements, each of which is associated with a
+different representation of the raw visual input. In fact,
+it was found recently that the features learnt in deep architectures resemble
+those observed in the first two of these stages (in areas V1 and V2
+of visual cortex) \citep{HonglakL2008}, and that they become more and
+more invariant to factors of variation (such as camera movement) in
+higher layers~\citep{Goodfellow2009}.
+It has been hypothesized that learning a hierarchy of features increases the
+ease and practicality of developing representations that are at once
+tailored to specific tasks, yet are able to borrow statistical strength
+from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
+feature representation can lead to higher-level (more abstract, more
+general) features that are more robust to unanticipated sources of
+variance extant in real data.
+Whereas a deep architecture can in principle be more powerful than a
+shallow one in terms of representation, depth appears to render the
+training problem more difficult in terms of optimization and local minima.
+It is also only recently that successful algorithms were proposed to
+overcome some of these difficulties.  All are based on unsupervised
+learning, often in an greedy layer-wise ``unsupervised pre-training''
+stage~\citep{Bengio-2009}.
+The principle is that each layer starting from
+the bottom is trained to represent its input (the output of the previous
+layer). After this
+unsupervised initialization, the stack of layers can be
+converted into a deep supervised feedforward neural network and fine-tuned by
+stochastic gradient descent.
+One of these layer initialization techniques,
+applied here, is the Denoising
+Auto-encoder~(DA)~\citep{VincentPLarochelleH2008-very-small} (see
+Figure~\ref{fig:da}), which performed similarly or
+better~\citep{VincentPLarochelleH2008-very-small} than previously
+proposed Restricted Boltzmann Machines (RBM)~\citep{Hinton06}
+in terms of unsupervised extraction
+of a hierarchy of features useful for classification. Each layer is trained
+to denoise its input, creating a layer of features that can be used as
+input for the next layer, forming a Stacked Denoising Auto-encoder (SDA).
+Note that training a Denoising Auto-encoder
+can actually been seen as training a particular RBM by an inductive
+principle different from maximum likelihood~\citep{Vincent-SM-2010},
+namely by Score Matching~\citep{Hyvarinen-2005,HyvarinenA2008}.
+Previous comparative experimental results with stacking of RBMs and DAs
+to build deep supervised predictors had shown that they could outperform
+shallow architectures in a variety of settings, especially
+when the data involves complex interactions between many factors of
+variation~\citep{LarochelleH2007,Bengio-2009}. Other experiments have suggested
+that the unsupervised layer-wise pre-training acted as a useful
+prior~\citep{Erhan+al-2010} that allows one to initialize a deep
+neural network in a relatively much smaller region of parameter space,
+corresponding to better generalization.
+To further the understanding of the reasons for the good performance
+observed with deep learners, we focus here on the following {\em hypothesis}:
+intermediate levels of representation, especially when there are
+more such levels, can be exploited to {\bf share
+statistical strength across different but related types of examples},
+such as examples coming from other tasks than the task of interest
+(the multi-task setting), or examples coming from an overlapping
+but different distribution (images with different kinds of perturbations
+and noises, here). This is consistent with the hypotheses discussed
+in~\citet{Bengio-2009} regarding the potential advantage
+of deep learning and the idea that more levels of representation can
+give rise to more abstract, more general features of the raw input.
+This hypothesis is related to a learning setting called
+{\bf self-taught learning}~\citep{RainaR2007}, which combines principles
+of semi-supervised and multi-task learning: the learner can exploit examples
+that are unlabeled and possibly come from a distribution different from the target
+distribution, e.g., from other classes than those of interest.
+It has already been shown that deep learners can clearly take advantage of
+unsupervised learning and unlabeled examples~\citep{Bengio-2009,WestonJ2008-small},
+but more needed to be done to explore the impact
+of {\em out-of-distribution} examples and of the {\em multi-task} setting
+(one exception is~\citep{CollobertR2008}, which shares and uses unsupervised
+pre-training only with the first layer). In particular the {\em relative
+advantage of deep learning} for these settings has not been evaluated.
+%
+The {\bf main claim} of this paper is that deep learners (with several levels of representation) can
+{\bf benefit more from out-of-distribution examples than shallow learners} (with a single
+level), both in the context of the multi-task setting and from
+perturbed examples. Because we are able to improve on state-of-the-art
+performance and reach human-level performance
+on a large-scale task, we consider that this paper is also a contribution
+to advance the application of machine learning to handwritten character recognition.
+More precisely, we ask and answer the following questions:
+%\begin{enumerate}
+$\bullet$ %\item
+Do the good results previously obtained with deep architectures on the
+MNIST digit images generalize to the setting of a similar but much larger and richer
+dataset, the NIST special database 19, with 62 classes and around 800k examples?
+$\bullet$ %\item
+To what extent does the perturbation of input images (e.g. adding
+noise, affine transformations, background images) make the resulting
+classifiers better not only on similarly perturbed images but also on
+the {\em original clean examples}? We study this question in the
+context of the 62-class and 10-class tasks of the NIST special database 19.
+$\bullet$ %\item
+Do deep architectures {\em benefit {\bf more} from such out-of-distribution}
+examples, in particular do they benefit more from
+examples that are perturbed versions of the examples from the task of interest?
+$\bullet$ %\item
+Similarly, does the feature learning step in deep learning algorithms benefit {\bf more}
+from training with moderately {\em different classes} (i.e. a multi-task learning scenario) than
+a corresponding shallow and purely supervised architecture?
+We train on 62 classes and test on 10 (digits) or 26 (upper case or lower case)
+to answer this question.
+%\end{enumerate}
+Our experimental results provide positive evidence towards all of these questions,
+as well as {\bf classifiers that reach human-level performance on 62-class isolated character
+recognition and beat previously published results on the NIST dataset (special database 19)}.
+To achieve these results, we introduce in the next section a sophisticated system
+for stochastically transforming character images and then explain the methodology,
+which is based on training with or without these transformed images and testing on
+clean ones.
+Code for generating these transformations as well as for the deep learning
+algorithms are made available at {\tt http://anonymous.url.net}.%{\tt http://hg.assembla.com/ift6266}.
+%\vspace*{-3mm}
+%\newpage
+\section{Perturbed and Transformed Character Images}
+\label{s:perturbations}
+%\vspace*{-2mm}
+Figure~\ref{fig:transform} shows the different transformations we used to stochastically
+transform $32 \times 32$ source images (such as the one in Fig.\ref{fig:torig})
+in order to obtain data from a larger distribution which
+covers a domain substantially larger than the clean characters distribution from
+which we start.
+Although character transformations have been used before to
+improve character recognizers, this effort is on a large scale both
+in number of classes and in the complexity of the transformations, hence
+in the complexity of the learning task.
+The code for these transformations (mostly Python) is available at
+{\tt http://anonymous.url.net}. All the modules in the pipeline (Figure~\ref{fig:transform}) share
+a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
+amount of deformation or noise introduced.
+There are two main parts in the pipeline. The first one,
+from thickness to pinch, performs transformations. The second
+part, from blur to contrast, adds different kinds of noise.
+More details can be found in~\citep{ift6266-tr-anonymous}.
+\begin{figure*}[ht]
+\centering
+\subfigure[Original]{\includegraphics[scale=0.6]{images/Original.png}\label{fig:torig}}
+\subfigure[Thickness]{\includegraphics[scale=0.6]{images/Thick_only.png}}
+\subfigure[Slant]{\includegraphics[scale=0.6]{images/Slant_only.png}}
+\subfigure[Affine Transformation]{\includegraphics[scale=0.6]{images/Affine_only.png}}
+\subfigure[Local Elastic Deformation]{\includegraphics[scale=0.6]{images/Localelasticdistorsions_only.png}}
+\subfigure[Pinch]{\includegraphics[scale=0.6]{images/Pinch_only.png}}
+%Noise
+\subfigure[Motion Blur]{\includegraphics[scale=0.6]{images/Motionblur_only.png}}
+\subfigure[Occlusion]{\includegraphics[scale=0.6]{images/occlusion_only.png}}
+\subfigure[Gaussian Smoothing]{\includegraphics[scale=0.6]{images/Bruitgauss_only.png}}
+\subfigure[Pixels Permutation]{\includegraphics[scale=0.6]{images/Permutpixel_only.png}}
+\subfigure[Gaussian Noise]{\includegraphics[scale=0.6]{images/Distorsiongauss_only.png}}
+\subfigure[Background Image Addition]{\includegraphics[scale=0.6]{images/background_other_only.png}}
+\subfigure[Salt \& Pepper]{\includegraphics[scale=0.6]{images/Poivresel_only.png}}
+\subfigure[Scratches]{\includegraphics[scale=0.6]{images/Rature_only.png}}
+\subfigure[Grey Level \& Contrast]{\includegraphics[scale=0.6]{images/Contrast_only.png}}
+\caption{Top left (a): example original image. Others (b-o): examples of the effect
+of each transformation module taken separately. Actual perturbed examples are obtained by
+a pipeline of these, with random choices about which module to apply and how much perturbation
+to apply.}
+\label{fig:transform}
+%\vspace*{-2mm}
+\end{figure*}
+%\vspace*{-3mm}
+\section{Experimental Setup}
+%\vspace*{-1mm}
+Much previous work on deep learning had been performed on
+the MNIST digits task~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006,Salakhutdinov+Hinton-2009},
+with 60,000 examples, and variants involving 10,000
+examples~\citep{Larochelle-jmlr-2009,VincentPLarochelleH2008-very-small}.
+The focus here is on much larger training sets, from 10 times to
+to 1000 times larger, and 62 classes.
+The first step in constructing the larger datasets (called NISTP and P07) is to sample from
+a {\em data source}: {\bf NIST} (NIST database 19), {\bf Fonts}, {\bf Captchas},
+and {\bf OCR data} (scanned machine printed characters). See more in
+Section~\ref{sec:sources} below. Once a character
+is sampled from one of these sources (chosen randomly), the second step is to
+apply a pipeline of transformations and/or noise processes outlined in section \ref{s:perturbations}.
+To provide a baseline of error rate comparison we also estimate human performance
+on both the 62-class task and the 10-class digits task.
+We compare the best Multi-Layer Perceptrons (MLP) against
+the best Stacked Denoising Auto-encoders (SDA), when
+both models' hyper-parameters are selected to minimize the validation set error.
+We also provide a comparison against a precise estimate
+of human performance obtained via Amazon's Mechanical Turk (AMT)
+service ({\tt http://mturk.com}).
+AMT users are paid small amounts
+of money to perform tasks for which human intelligence is required.
+Mechanical Turk has been used extensively in natural language processing and vision.
+%processing \citep{SnowEtAl2008} and vision
+%\citep{SorokinAndForsyth2008,whitehill09}.
+AMT users were presented
+with 10 character images (from a test set) on a screen
+and asked to label them.
+They were forced to choose a single character class (either among the
+62 or 10 character classes) for each image.
+80 subjects classified 2500 images per (dataset,task) pair.
+Different humans labelers sometimes provided a different label for the same
+example, and we were able to estimate the error variance due to this effect
+because each image was classified by 3 different persons.
+The average error of humans on the 62-class task NIST test set
+is 18.2\%, with a standard error of 0.1\%.
+We controlled noise in the labelling process by (1)
+requiring AMT workers with a higher than normal average of accepted
+responses ($>$95\%) on other tasks (2) discarding responses that were not
+complete (10 predictions) (3) discarding responses for which for which the
+time to predict was smaller than 3 seconds for NIST (the mean response time
+was 20 seconds) and 6 seconds seconds for NISTP (average response time of
+45 seconds) (4) discarding responses which were obviously wrong (10
+identical ones, or "12345..."). Overall, after such filtering, we kept
+approximately 95\% of the AMT workers' responses.
+%\vspace*{-3mm}
+\subsection{Data Sources}
+\label{sec:sources}
+%\vspace*{-2mm}
+%\begin{itemize}
+%\item
+{\bf NIST.}
+Our main source of characters is the NIST Special Database 19~\citep{Grother-1995},
+widely used for training and testing character
+recognition systems~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}.
+The dataset is composed of 814255 digits and characters (upper and lower cases), with hand checked classifications,
+extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes
+corresponding to ``0''-``9'',``A''-``Z'' and ``a''-``z''. The dataset contains 8 parts (partitions) of varying complexity.
+The fourth partition (called $hsf_4$, 82,587 examples),
+experimentally recognized to be the most difficult one, is the one recommended
+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}
+for that purpose. We randomly split the remainder (731,668 examples) into a training set and a validation set for
+model selection.
+The performances reported by previous work on that dataset mostly use only the digits.
+Here we use all the classes both in the training and testing phase. This is especially
+useful to estimate the effect of a multi-task setting.
+The distribution of the classes in the NIST training and test sets differs
+substantially, with relatively many more digits in the test set, and a more uniform distribution
+of letters in the test set (whereas in the training set they are distributed
+more like in natural text).
+%\vspace*{-1mm}
+%\item
+{\bf Fonts.}
+In order to have a good variety of sources we downloaded an important number of free fonts from:
+{\tt http://cg.scs.carleton.ca/\textasciitilde luc/freefonts.html}.
+% TODO: pointless to anonymize, it's not pointing to our work
+Including an operating system's (Windows 7) fonts, there is a total of $9817$ different fonts that we can choose uniformly from.
+The chosen {\tt ttf} file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image,
+directly as input to our models.
+%\vspace*{-1mm}
+%\item
+{\bf Captchas.}
+The Captcha data source is an adaptation of the \emph{pycaptcha} library (a Python-based captcha generator library) for
+generating characters of the same format as the NIST dataset. This software is based on
+a random character class generator and various kinds of transformations similar to those described in the previous sections.
+In order to increase the variability of the data generated, many different fonts are used for generating the characters.
+Transformations (slant, distortions, rotation, translation) are applied to each randomly generated character with a complexity
+depending on the value of the complexity parameter provided by the user of the data source.
+%Two levels of complexity are allowed and can be controlled via an easy to use facade class. %TODO: what's a facade class?
+%\vspace*{-1mm}
+%\item
+{\bf OCR data.}
+A large set (2 million) of scanned, OCRed and manually verified machine-printed
+characters where included as an
+additional source. This set is part of a larger corpus being collected by the Image Understanding
+Pattern Recognition Research group led by Thomas Breuel at University of Kaiserslautern
+({\tt http://www.iupr.com}), and which will be publicly released.
+%TODO: let's hope that Thomas is not a reviewer! :) Seriously though, maybe we should anonymize this
+%\end{itemize}
+%\vspace*{-3mm}
+\subsection{Data Sets}
+%\vspace*{-2mm}
+All data sets contain 32$\times$32 grey-level images (values in $[0,1]$) associated with a label
+from one of the 62 character classes.
+%\begin{itemize}
+%\vspace*{-1mm}
+%\item
+{\bf NIST.} This is the raw NIST special database 19~\citep{Grother-1995}. It has
+\{651,668 / 80,000 / 82,587\} \{training / validation / test\} examples.
+%\vspace*{-1mm}
+%\item
+{\bf P07.} This dataset is obtained by taking raw characters from all four of the above sources
+and sending them through the transformation pipeline described in section \ref{s:perturbations}.
+For each new example to generate, a data source is selected with probability $10\%$ from the fonts,
+$25\%$ from the captchas, $25\%$ from the OCR data and $40\%$ from NIST. We apply all the transformations in the
+order given above, and for each of them we sample uniformly a \emph{complexity} in the range $[0,0.7]$.
+It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples
+obtained from the corresponding NIST sets plus other sources.
+%\vspace*{-1mm}
+%\item
+{\bf NISTP.} This one is equivalent to P07 (complexity parameter of $0.7$ with the same proportions of data sources)
+except that we only apply
+transformations from slant to pinch (see Fig.\ref{fig:transform}(b-f)).
+Therefore, the character is
+transformed but no additional noise is added to the image, giving images
+closer to the NIST dataset.
+It has \{81,920,000 / 80,000 / 20,000\} \{training / validation / test\} examples
+obtained from the corresponding NIST sets plus other sources.
+%\end{itemize}
+\begin{figure*}[ht]
+%\vspace*{-2mm}
+\centerline{\resizebox{0.8\textwidth}{!}{\includegraphics{images/denoising_autoencoder_small.pdf}}}
+%\vspace*{-2mm}
+\caption{Illustration of the computations and training criterion for the denoising
+auto-encoder used to pre-train each layer of the deep architecture. Input $x$ of
+the layer (i.e. raw input or output of previous layer)
+s corrupted into $\tilde{x}$ and encoded into code $y$ by the encoder $f_\theta(\cdot)$.
+The decoder $g_{\theta'}(\cdot)$ maps $y$ to reconstruction $z$, which
+is compared to the uncorrupted input $x$ through the loss function
+$L_H(x,z)$, whose expected value is approximately minimized during training
+by tuning $\theta$ and $\theta'$.}
+\label{fig:da}
+%\vspace*{-2mm}
+\end{figure*}
+%\vspace*{-3mm}
+\subsection{Models and their Hyper-parameters}
+%\vspace*{-2mm}
+The experiments are performed using MLPs (with a single
+hidden layer) and deep SDAs.
+\emph{Hyper-parameters are selected based on the {\bf NISTP} validation set error.}
+{\bf Multi-Layer Perceptrons (MLP).}  Whereas previous work had compared
+deep architectures to both shallow MLPs and SVMs, we only compared to MLPs
+here because of the very large datasets used (making the use of SVMs
+computationally challenging because of their quadratic scaling
+behavior). Preliminary experiments on training SVMs (libSVM) with subsets
+of the training set allowing the program to fit in memory yielded
+substantially worse results than those obtained with MLPs\footnote{RBF SVMs
+trained with a subset of NISTP or NIST, 100k examples, to fit in memory,
+yielded 64\% test error or worse; online linear SVMs trained on the whole
+of NIST or 800k from NISTP yielded no better than 42\% error; slightly
+better results were obtained by sparsifying the pixel intensities and
+projecting to a second-order polynomial (a very sparse vector), still
+41\% error. We expect that better results could be obtained with a
+better implementation allowing for training with more examples and
+a higher-order non-linear projection.}  For training on nearly a hundred million examples (with the
+perturbed data), the MLPs and SDA are much more convenient than classifiers
+based on kernel methods.  The MLP has a single hidden layer with $\tanh$
+activation functions, and softmax (normalized exponentials) on the output
+layer for estimating $P(class | image)$.  The number of hidden units is
+taken in $\{300,500,800,1000,1500\}$.  Training examples are presented in
+minibatches of size 20. A constant learning rate was chosen among $\{0.001,
+0.01, 0.025, 0.075, 0.1, 0.5\}$.
+%through preliminary experiments (measuring performance on a validation set),
+%and $0.1$ (which was found to work best) was then selected for optimizing on
+%the whole training sets.
+%\vspace*{-1mm}
+{\bf Stacked Denoising Auto-encoders (SDA).}
+Various auto-encoder variants and Restricted Boltzmann Machines (RBMs)
+can be used to initialize the weights of each layer of a deep MLP (with many hidden
+layers)~\citep{Hinton06,ranzato-07-small,Bengio-nips-2006},
+apparently setting parameters in the
+basin of attraction of supervised gradient descent yielding better
+generalization~\citep{Erhan+al-2010}.  This initial {\em unsupervised
+pre-training phase} uses all of the training images but not the training labels.
+Each layer is trained in turn to produce a new representation of its input
+(starting from the raw pixels).
+It is hypothesized that the
+advantage brought by this procedure stems from a better prior,
+on the one hand taking advantage of the link between the input
+distribution $P(x)$ and the conditional distribution of interest
+$P(y|x)$ (like in semi-supervised learning), and on the other hand
+taking advantage of the expressive power and bias implicit in the
+deep architecture (whereby complex concepts are expressed as
+compositions of simpler ones through a deep hierarchy).
+Here we chose to use the Denoising
+Auto-encoder~\citep{VincentPLarochelleH2008-very-small} as the building block for
+these deep hierarchies of features, as it is simple to train and
+explain (see Figure~\ref{fig:da}, as well as
+tutorial and code there: {\tt http://deeplearning.net/tutorial}),
+provides efficient inference, and yielded results
+comparable or better than RBMs in series of experiments
+\citep{VincentPLarochelleH2008-very-small}. It really corresponds to a Gaussian
+RBM trained by a Score Matching criterion~\cite{Vincent-SM-2010}.
+During training, a Denoising
+Auto-encoder is presented with a stochastically corrupted version
+of the input and trained to reconstruct the uncorrupted input,
+forcing the hidden units to represent the leading regularities in
+the data. Here we use the random binary masking corruption
+(which sets to 0 a random subset of the inputs).
+Once it is trained, in a purely unsupervised way,
+its hidden units' activations can
+be used as inputs for training a second one, etc.
+After this unsupervised pre-training stage, the parameters
+are used to initialize a deep MLP, which is fine-tuned by
+the same standard procedure used to train them (see above).
+The SDA hyper-parameters are the same as for the MLP, with the addition of the
+amount of corruption noise (we used the masking noise process, whereby a
+fixed proportion of the input values, randomly selected, are zeroed), and a
+separate learning rate for the unsupervised pre-training stage (selected
+from the same above set). The fraction of inputs corrupted was selected
+among $\{10\%, 20\%, 50\%\}$. Another hyper-parameter is the number
+of hidden layers but it was fixed to 3 for most experiments,
+based on previous work with
+SDAs on MNIST~\citep{VincentPLarochelleH2008-very-small}.
+We also compared against 1 and against 2 hidden layers, in order
+to disantangle the effect of depth from the effect of unsupervised
+pre-training.
+The size of the hidden
+layers was kept constant across hidden layers, and the best results
+were obtained with the largest values that we could experiment
+with given our patience, with 1000 hidden units.
+%\vspace*{-1mm}
+\begin{figure*}[ht]
+%\vspace*{-2mm}
+\centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}}
+%\vspace*{-3mm}
+\caption{SDAx are the {\bf deep} models. Error bars indicate a 95\% confidence interval. 0 indicates that the model was trained
+on NIST, 1 on NISTP, and 2 on P07. Left: overall results
+of all models, on NIST and NISTP test sets.
+Right: error rates on NIST test digits only, along with the previous results from
+literature~\citep{Granger+al-2007,Cortes+al-2000,Oliveira+al-2002-short,Milgram+al-2005}
+respectively based on ART, nearest neighbors, MLPs, and SVMs.}
+\label{fig:error-rates-charts}
+%\vspace*{-2mm}
+\end{figure*}
+\begin{figure*}[ht]
+\vspace*{-3mm}
+\centerline{\resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}}
+\vspace*{-3mm}
+\caption{Relative improvement in error rate due to out-of-distribution examples.
+Left: Improvement (or loss, when negative)
+induced by out-of-distribution examples (perturbed data).
+Right: Improvement (or loss, when negative) induced by multi-task
+learning (training on all classes and testing only on either digits,
+upper case, or lower-case). The deep learner (SDA) benefits more from
+out-of-distribution examples, compared to the shallow MLP.}
+\label{fig:improvements-charts}
+\vspace*{-2mm}
+\end{figure*}
+\vspace*{-2mm}
+\section{Experimental Results}
+\vspace*{-2mm}
+%%\vspace*{-1mm}
+%\subsection{SDA vs MLP vs Humans}
+%%\vspace*{-1mm}
+The models are either trained on NIST (MLP0 and SDA0),
+NISTP (MLP1 and SDA1), or P07 (MLP2 and SDA2), and tested
+on either NIST, NISTP or P07 (regardless of the data set used for training),
+either on the 62-class task
+or on the 10-digits task. Training time (including about half
+for unsupervised pre-training, for DAs) on the larger
+datasets is around one day on a GPU (GTX 285).
+Figure~\ref{fig:error-rates-charts} summarizes the results obtained,
+comparing humans, the three MLPs (MLP0, MLP1, MLP2) and the three SDAs (SDA0, SDA1,
+SDA2), along with the previous results on the digits NIST special database
+19 test set from the literature, respectively based on ARTMAP neural
+networks ~\citep{Granger+al-2007}, fast nearest-neighbor search
+~\citep{Cortes+al-2000}, MLPs ~\citep{Oliveira+al-2002-short}, and SVMs
+~\citep{Milgram+al-2005}.%  More detailed and complete numerical results
+%(figures and tables, including standard errors on the error rates) can be
+%found in Appendix.
+The deep learner not only outperformed the shallow ones and
+previously published performance (in a statistically and qualitatively
+significant way) but when trained with perturbed data
+reaches human performance on both the 62-class task
+and the 10-class (digits) task.
+17\% error (SDA1) or 18\% error (humans) may seem large but a large
+majority of the errors from humans and from SDA1 are from out-of-context
+confusions (e.g. a vertical bar can be a ``1'', an ``l'' or an ``L'', and a
+``c'' and a ``C'' are often indistinguishible).
+Regarding shallower networks pre-trained with unsupervised denoising
+auto-encders, we find that the NIST test error is 21\% with one hidden
+layer and 20\% with two hidden layers (vs 17\% in the same conditions
+with 3 hidden layers). Compare this with the 23\% error achieved
+by the MLP, i.e. a single hidden layer and no unsupervised pre-training.
+As found in previous work~\cite{Erhan+al-2010,Larochelle-jmlr-2009},
+these results show that both depth and
+unsupervised pre-training need to be combined in order to achieve
+the best results.
+In addition, as shown in the left of
+Figure~\ref{fig:improvements-charts}, the relative improvement in error
+rate brought by out-of-distribution examples is greater for the deep
+SDA, and these
+differences with the shallow MLP are statistically and qualitatively
+significant.
+The left side of the figure shows the improvement to the clean
+NIST test set error brought by the use of out-of-distribution examples
+(i.e. the perturbed examples examples from NISTP or P07),
+over the models trained exclusively on NIST (respectively SDA0 and MLP0).
+Relative percent change is measured by taking
+$100 \% \times$ (original model's error / perturbed-data model's error - 1).
+The right side of
+Figure~\ref{fig:improvements-charts} shows the relative improvement
+brought by the use of a multi-task setting, in which the same model is
+trained for more classes than the target classes of interest (i.e. training
+with all 62 classes when the target classes are respectively the digits,
+lower-case, or upper-case characters). Again, whereas the gain from the
+multi-task setting is marginal or negative for the MLP, it is substantial
+for the SDA.  Note that to simplify these multi-task experiments, only the original
+NIST dataset is used. For example, the MLP-digits bar shows the relative
+percent improvement in MLP error rate on the NIST digits test set
+as $100\% \times$ (single-task
+model's error / multi-task model's error - 1).  The single-task model is
+trained with only 10 outputs (one per digit), seeing only digit examples,
+whereas the multi-task model is trained with 62 outputs, with all 62
+character classes as examples.  Hence the hidden units are shared across
+all tasks.  For the multi-task model, the digit error rate is measured by
+comparing the correct digit class with the output class associated with the
+maximum conditional probability among only the digit classes outputs.  The
+setting is similar for the other two target classes (lower case characters
+and upper case characters). Note however that some types of perturbations
+(NISTP) help more than others (P07) when testing on the clean images.
+%%\vspace*{-1mm}
+%\subsection{Perturbed Training Data More Helpful for SDA}
+%%\vspace*{-1mm}
+%%\vspace*{-1mm}
+%\subsection{Multi-Task Learning Effects}
+%%\vspace*{-1mm}
+\iffalse
+As previously seen, the SDA is better able to benefit from the
+transformations applied to the data than the MLP. In this experiment we
+define three tasks: recognizing digits (knowing that the input is a digit),
+recognizing upper case characters (knowing that the input is one), and
+recognizing lower case characters (knowing that the input is one).  We
+consider the digit classification task as the target task and we want to
+evaluate whether training with the other tasks can help or hurt, and
+whether the effect is different for MLPs versus SDAs.  The goal is to find
+out if deep learning can benefit more (or less) from multiple related tasks
+(i.e. the multi-task setting) compared to a corresponding purely supervised
+shallow learner.
+We use a single hidden layer MLP with 1000 hidden units, and a SDA
+with 3 hidden layers (1000 hidden units per layer), pre-trained and
+fine-tuned on NIST.
+Our results show that the MLP benefits marginally from the multi-task setting
+in the case of digits (5\% relative improvement) but is actually hurt in the case
+of characters (respectively 3\% and 4\% worse for lower and upper class characters).
+On the other hand the SDA benefited from the multi-task setting, with relative
+error rate improvements of 27\%, 15\% and 13\% respectively for digits,
+lower and upper case characters, as shown in Table~\ref{tab:multi-task}.
+\fi
+\vspace*{-2mm}
+\section{Conclusions and Discussion}
+\vspace*{-2mm}
+We have found that out-of-distribution examples (multi-task learning
+and perturbed examples) are more beneficial
+to a deep learner than to a traditional shallow and purely
+supervised learner. More precisely,
+the answers are positive for all the questions asked in the introduction.
+%\begin{itemize}
+$\bullet$ %\item
+{\bf Do the good results previously obtained with deep architectures on the
+MNIST digits generalize to a much larger and richer (but similar)
+dataset, the NIST special database 19, with 62 classes and around 800k examples}?
+Yes, the SDA {\em systematically outperformed the MLP and all the previously
+published results on this dataset} (the ones that we are aware of), {\em in fact reaching human-level
+performance} at around 17\% error on the 62-class task and 1.4\% on the digits,
+and beating previously published results on the same data.
+$\bullet$ %\item
+{\bf To what extent do out-of-distribution examples help deep learners,
+and do they help them more than shallow supervised ones}?
+We found that distorted training examples not only made the resulting
+classifier better on similarly perturbed images but also on
+the {\em original clean examples}, and more importantly and more novel,
+that deep architectures benefit more from such {\em out-of-distribution}
+examples. Shallow MLPs were helped by perturbed training examples when tested on perturbed input
+images (65\% relative improvement on NISTP)
+but only marginally helped (5\% relative improvement on all classes)
+or even hurt (10\% relative loss on digits)
+with respect to clean examples. On the other hand, the deep SDAs
+were significantly boosted by these out-of-distribution examples.
+Similarly, whereas the improvement due to the multi-task setting was marginal or
+negative for the MLP (from +5.6\% to -3.6\% relative change),
+it was quite significant for the SDA (from +13\% to +27\% relative change),
+which may be explained by the arguments below.
+Since out-of-distribution data
+(perturbed or from other related classes) is very common, this conclusion
+is of practical importance.
+%\end{itemize}
+In the original self-taught learning framework~\citep{RainaR2007}, the
+out-of-sample examples were used as a source of unsupervised data, and
+experiments showed its positive effects in a \emph{limited labeled data}
+scenario. However, many of the results by \citet{RainaR2007} (who used a
+shallow, sparse coding approach) suggest that the {\em relative gain of self-taught
+learning vs ordinary supervised learning} diminishes as the number of labeled examples increases.
+We note instead that, for deep
+architectures, our experiments show that such a positive effect is accomplished
+even in a scenario with a \emph{large number of labeled examples},
+i.e., here, the relative gain of self-taught learning and
+out-of-distribution examples is probably preserved
+in the asymptotic regime. However, note that in our perturbation experiments
+(but not in our multi-task experiments),
+even the out-of-distribution examples are labeled, unlike in the
+earlier self-taught learning experiments~\citep{RainaR2007}.
+{\bf Why would deep learners benefit more from the self-taught learning
+framework and out-of-distribution examples}?
+The key idea is that the lower layers of the predictor compute a hierarchy
+of features that can be shared across tasks or across variants of the
+input distribution. A theoretical analysis of generalization improvements
+due to sharing of intermediate features across tasks already points
+towards that explanation~\cite{baxter95a}.
+Intermediate features that can be used in different
+contexts can be estimated in a way that allows to share statistical
+strength. Features extracted through many levels are more likely to
+be more abstract and more invariant to some of the factors of variation
+in the underlying distribution (as the experiments in~\citet{Goodfellow2009} suggest),
+increasing the likelihood that they would be useful for a larger array
+of tasks and input conditions.
+Therefore, we hypothesize that both depth and unsupervised
+pre-training play a part in explaining the advantages observed here, and future
+experiments could attempt at teasing apart these factors.
+And why would deep learners benefit from the self-taught learning
+scenarios even when the number of labeled examples is very large?
+We hypothesize that this is related to the hypotheses studied
+in~\citet{Erhan+al-2010}. In~\citet{Erhan+al-2010}
+it was found that online learning on a huge dataset did not make the
+advantage of the deep learning bias vanish, and a similar phenomenon
+may be happening here. We hypothesize that unsupervised pre-training
+of a deep hierarchy with out-of-distribution examples initializes the
+model in the basin of attraction of supervised gradient descent
+that corresponds to better generalization. Furthermore, such good
+basins of attraction are not discovered by pure supervised learning
+(with or without out-of-distribution examples) from random initialization, and more labeled examples
+does not allow the shallow or purely supervised models to discover
+the kind of better basins associated
+with deep learning and out-of-distribution examples.
+A Flash demo of the recognizer (where both the MLP and the SDA can be compared)
+can be executed on-line at the anonymous site {\tt http://deep.host22.com}.
+\iffalse
+\section*{Appendix I: Detailed Numerical Results}
+These tables correspond to Figures 2 and 3 and contain the raw error rates for each model and dataset considered.
+They also contain additional data such as test errors on P07 and standard errors.
+\begin{table}[ht]
+\caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits +
+26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training
+(SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture
+(MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07)
+and using a validation set to select hyper-parameters and other training choices.
+\{SDA,MLP\}0 are trained on NIST,
+\{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07.
+The human error rate on digits is a lower bound because it does not count digits that were
+recognized as letters. For comparison, the results found in the literature
+on NIST digits classification using the same test set are included.}
+\label{tab:sda-vs-mlp-vs-humans}
+\begin{center}
+\begin{tabular}{|l|r|r|r|r|} \hline
+& NIST test          & NISTP test       & P07 test       & NIST test digits   \\ \hline
+Humans&   18.2\% $\pm$.1\%   &  39.4\%$\pm$.1\%   &  46.9\%$\pm$.1\%  &  $1.4\%$ \\ \hline
+SDA0   &  23.7\% $\pm$.14\%  &  65.2\%$\pm$.34\%  & 97.45\%$\pm$.06\%  & 2.7\% $\pm$.14\%\\ \hline
+SDA1   &  17.1\% $\pm$.13\%  &  29.7\%$\pm$.3\%  & 29.7\%$\pm$.3\%  & 1.4\% $\pm$.1\%\\ \hline
+SDA2   &  18.7\% $\pm$.13\%  &  33.6\%$\pm$.3\%  & 39.9\%$\pm$.17\%  & 1.7\% $\pm$.1\%\\ \hline
+MLP0   &  24.2\% $\pm$.15\%  & 68.8\%$\pm$.33\%  & 78.70\%$\pm$.14\%  & 3.45\% $\pm$.15\% \\ \hline
+MLP1   &  23.0\% $\pm$.15\%  &  41.8\%$\pm$.35\%  & 90.4\%$\pm$.1\%  & 3.85\% $\pm$.16\% \\ \hline
+MLP2   &  24.3\% $\pm$.15\%  &  46.0\%$\pm$.35\%  & 54.7\%$\pm$.17\%  & 4.85\% $\pm$.18\% \\ \hline
+\citep{Granger+al-2007} &     &                    &                   & 4.95\% $\pm$.18\% \\ \hline
+\citep{Cortes+al-2000} &      &                    &                   & 3.71\% $\pm$.16\% \\ \hline
+\citep{Oliveira+al-2002} &    &                    &                   & 2.4\% $\pm$.13\% \\ \hline
+\citep{Milgram+al-2005} &      &                    &                   & 2.1\% $\pm$.12\% \\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+\begin{table}[ht]
+\caption{Relative change in error rates due to the use of perturbed training data,
+either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models.
+A positive value indicates that training on the perturbed data helped for the
+given test set (the first 3 columns on the 62-class tasks and the last one is
+on the clean 10-class digits). Clearly, the deep learning models did benefit more
+from perturbed training data, even when testing on clean data, whereas the MLP
+trained on perturbed data performed worse on the clean digits and about the same
+on the clean characters. }
+\label{tab:perturbation-effect}
+\begin{center}
+\begin{tabular}{|l|r|r|r|r|} \hline
+& NIST test          & NISTP test      & P07 test       & NIST test digits   \\ \hline
+SDA0/SDA1-1   &  38\%      &  84\%           & 228\%          &  93\% \\ \hline
+SDA0/SDA2-1   &  27\%      &  94\%           & 144\%          &  59\% \\ \hline
+MLP0/MLP1-1   &  5.2\%     &  65\%           & -13\%          & -10\%  \\ \hline
+MLP0/MLP2-1   &  -0.4\%    &  49\%           & 44\%           & -29\% \\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+\begin{table}[ht]
+\caption{Test error rates and relative change in error rates due to the use of
+a multi-task setting, i.e., training on each task in isolation vs training
+for all three tasks together, for MLPs vs SDAs. The SDA benefits much
+more from the multi-task setting. All experiments on only on the
+unperturbed NIST data, using validation error for model selection.
+Relative improvement is 1 - single-task error / multi-task error.}
+\label{tab:multi-task}
+\begin{center}
+\begin{tabular}{|l|r|r|r|} \hline
+& single-task  & multi-task  & relative \\
+& setting      & setting     & improvement \\ \hline
+MLP-digits   &  3.77\%      &  3.99\%     & 5.6\%   \\ \hline
+MLP-lower   &  17.4\%      &  16.8\%     &  -4.1\%    \\ \hline
+MLP-upper   &  7.84\%     &  7.54\%      & -3.6\%    \\ \hline
+SDA-digits   &  2.6\%      &  3.56\%     & 27\%    \\ \hline
+SDA-lower   &  12.3\%      &  14.4\%    & 15\%    \\ \hline
+SDA-upper   &  5.93\%     &  6.78\%      & 13\%    \\ \hline
+\end{tabular}
+\end{center}
+\end{table}
+\fi
+%\afterpage{\clearpage}
+%\clearpage
+{
+%\bibliographystyle{spbasic}      % basic style, author-year citations
+\bibliographystyle{plainnat}
+\bibliography{strings,strings-short,strings-shorter,ift6266_ml,specials,aigaion-shorter}
+%\bibliographystyle{unsrtnat}
+%\bibliographystyle{apalike}
+}
+\end{document}

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