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\twocolumn[
\aistatstitle{Deep Learners Benefit More from Out-of-Distribution Examples}
\runningtitle{Deep Learners for Out-of-Distribution Examples}
\runningauthor{Bengio et. al.}
\aistatsauthor{
Yoshua  Bengio \and
Frédéric  Bastien \and
\bf Arnaud  Bergeron \and
Nicolas  Boulanger-Lewandowski \and \\
\bf Thomas  Breuel \and
Youssouf  Chherawala \and
\bf Moustapha  Cisse \and 
Myriam  Côté \and  \\
\bf Dumitru  Erhan \and
Jeremy  Eustache \and
\bf Xavier  Glorot \and 
Xavier  Muller \and \\
\bf Sylvain  Pannetier Lebeuf \and
Razvan  Pascanu \and
\bf Salah  Rifai \and 
Francois  Savard \and  \\
\bf Guillaume  Sicard \\
\vspace*{1mm}}

%I can't use aistatsaddress in a single side paragraphe.
%The document is 2 colums, but this section span the 2 colums, sot there is only 1 left
\center{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada}
\vspace*{5mm}
]
%\aistatsaddress{Dept. IRO, U. Montreal, P.O. Box 6128, Centre-Ville branch, H3C 3J7, Montreal (Qc), Canada}


%\vspace*{5mm}}
%\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://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://hg.assembla.com/ift6266}. 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{ARXIV-2010}.

\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).}  The MLP output estimated with
\[
P({\rm class}|{\rm input}=x)={\rm softmax}(b_2+W_2\tanh(b_1+W_1 x)),
\] 
i.e., two layers, where 
\[
 p={\rm softmax}(a)
\]
means that 
\[
 p_i(x)=\exp(a_i)/\sum_j \exp(a_j)
\] 
representing the probability 
for class $i$, $\tanh$ is the element-wise
hyperbolic tangent, $b_i$ are parameter vectors, and $W_i$ are 
parameter matrices (one per layer). The
number of rows of $W_1$ is called the number of hidden units (of the
single hidden layer, here), and
is one way to control capacity (the main other ways to control capacity are
the number of training iterations and optionally a regularization penalty
on the parameters, not used here because it did not help).
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 $P({\rm class} | {\rm input})$.  The number of hidden units is
taken in $\{300,500,800,1000,1500\}$.  Training examples are presented in
minibatches of size 20, i.e., the parameters are iteratively updated in the direction
of the mean gradient of the next 20 examples. 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 its unsupervised training, a Denoising
Auto-encoder is presented with a stochastically corrupted version $\tilde{x}$
of the input $x$ and trained to reconstruct to produce a reconstruction $z$ 
of the uncorrupted input $x$. Because the network has to denoise, it is
forcing the hidden units $y$ to represent the leading regularities in
the data. Following~\citep{VincentPLarochelleH2008-very-small} 
the hidden units output $y$ is obtained through the sigmoid-affine
encoder
\[
 y={\rm sigm}(c+V x)
\]
where ${\rm sigm}(a)=1/(1+\exp(-a))$
and the reconstruction is obtained through the same transformation
\[ 
 z={\rm sigm}(d+V' y)
\]
using the transpose of encoder weights.
The training
set average of the cross-entropy
reconstruction loss 
\[
 L_H(x,z)=\sum_i z_i \log x_i + (1-z_i) \log(1-x_i)
\]
is minimized.
Here we use the random binary masking corruption
(which in $\tilde{x}$ sets to 0 a random subset of the elements of $x$, and
copies the rest).
Once the first denoising auto-encoder is trained, its parameters can be used
to set the first layer of the deep MLP. The original data are then processed
through that first layer, and the output of the hidden units form a new
representation that can be used as input data for training a second denoising
auto-encoder, still in a purely unsupervised way.
This is repeated for the desired number of hidden layers.
After this unsupervised pre-training stage, the parameters
are used to initialize a deep MLP (similar to the above, but
with more layers), which is fine-tuned by
the same standard procedure (stochastic gradient descent)
used to train MLPs in general (see above).
The top layer parameters of the deep MLP (the one which outputs the
class probabilities and takes the top hidden layer as input) can
be initialized at 0.
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 our experiments,
based on previous work with
SDAs on MNIST~\citep{VincentPLarochelleH2008-very-small}. 
We also compared against 1 and against 2 hidden layers, 
to disantangle the effect of depth from that of unsupervised
pre-training.
The size of each hidden
layer was kept constant across hidden layers, and the best results
were obtained with the largest values that we tried
(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 {\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

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