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Added charts to nips2010_submission.tex
author fsavard
date Sun, 30 May 2010 12:04:05 -0400
parents d02d288257bf
children db28764b8252
rev   line source
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
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1 \documentclass{article} % For LaTeX2e
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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2 \usepackage{nips10submit_e,times}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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3
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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4 \usepackage{amsthm,amsmath,amssymb,bbold,bbm}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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5 \usepackage{algorithm,algorithmic}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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6 \usepackage[utf8]{inputenc}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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7 \usepackage{graphicx,subfigure}
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8 \usepackage[numbers]{natbib}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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10 \title{Generating and Exploiting Perturbed and Multi-Task Handwritten Training Data for Deep Architectures}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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11 \author{The IFT6266 Gang}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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12
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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13 \begin{document}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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14
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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15 %\makeanontitle
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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16 \maketitle
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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17
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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18 \begin{abstract}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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19 Recent theoretical and empirical work in statistical machine learning has
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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20 demonstrated the importance of learning algorithms for deep
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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21 architectures, i.e., function classes obtained by composing multiple
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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22 non-linear transformations. In the area of handwriting recognition,
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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23 deep learning algorithms
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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24 had been evaluated on rather small datasets with a few tens of thousands
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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25 of examples. Here we propose a powerful generator of variations
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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26 of examples for character images based on a pipeline of stochastic
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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27 transformations that include not only the usual affine transformations
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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28 but also the addition of slant, local elastic deformations, changes
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
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29 in thickness, background images, color, contrast, occlusion, and
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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30 various types of pixel and spatially correlated noise.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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31 We evaluate a deep learning algorithm (Stacked Denoising Autoencoders)
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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32 on the task of learning to classify digits and letters transformed
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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33 with this pipeline, using the hundreds of millions of generated examples
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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34 and testing on the full 62-class NIST test set.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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35 We find that the SDA outperforms its
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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36 shallow counterpart, an ordinary Multi-Layer Perceptron,
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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37 and that it is better able to take advantage of the additional
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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38 generated data, as well as better able to take advantage of
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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39 the multi-task setting, i.e.,
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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40 training from more classes than those of interest in the end.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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41 In fact, we find that the SDA reaches human performance as
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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42 estimated by the Amazon Mechanical Turk on the 62-class NIST test characters.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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43 \end{abstract}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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44
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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45 \section{Introduction}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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46
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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47 Deep Learning has emerged as a promising new area of research in
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48 statistical machine learning (see~\citet{Bengio-2009} for a review).
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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49 Learning algorithms for deep architectures are centered on the learning
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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50 of useful representations of data, which are better suited to the task at hand.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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51 This is in great part inspired by observations of the mammalian visual cortex,
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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52 which consists of a chain of processing elements, each of which is associated with a
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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53 different representation. In fact,
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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54 it was found recently that the features learnt in deep architectures resemble
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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55 those observed in the first two of these stages (in areas V1 and V2
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56 of visual cortex)~\citep{HonglakL2008}.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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57 Processing images typically involves transforming the raw pixel data into
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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58 new {\bf representations} that can be used for analysis or classification.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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59 For example, a principal component analysis representation linearly projects
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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60 the input image into a lower-dimensional feature space.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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61 Why learn a representation? Current practice in the computer vision
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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62 literature converts the raw pixels into a hand-crafted representation
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63 e.g.\ SIFT features~\citep{Lowe04}, but deep learning algorithms
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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64 tend to discover similar features in their first few
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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65 levels~\citep{HonglakL2008,ranzato-08,Koray-08,VincentPLarochelleH2008-very-small}.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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66 Learning increases the
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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67 ease and practicality of developing representations that are at once
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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68 tailored to specific tasks, yet are able to borrow statistical strength
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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69 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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70 feature representation can lead to higher-level (more abstract, more
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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71 general) features that are more robust to unanticipated sources of
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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72 variance extant in real data.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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73
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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74 Whereas a deep architecture can in principle be more powerful than a
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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75 shallow one in terms of representation, depth appears to render the
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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76 training problem more difficult in terms of optimization and local minima.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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77 It is also only recently that successful algorithms were proposed to
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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78 overcome some of these difficulties. All are based on unsupervised
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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79 learning, often in an greedy layer-wise ``unsupervised pre-training''
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80 stage~\citep{Bengio-2009}. One of these layer initialization techniques,
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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81 applied here, is the Denoising
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82 Auto-Encoder~(DEA)~\citep{VincentPLarochelleH2008-very-small}, which
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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83 performed similarly or better than previously proposed Restricted Boltzmann
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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84 Machines in terms of unsupervised extraction of a hierarchy of features
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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85 useful for classification. The principle is that each layer starting from
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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86 the bottom is trained to encode their input (the output of the previous
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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87 layer) and try to reconstruct it from a corrupted version of it. After this
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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88 unsupervised initialization, the stack of denoising auto-encoders can be
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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89 converted into a deep supervised feedforward neural network and trained by
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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90 stochastic gradient descent.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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91
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92 In this paper we ask the following questions:
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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93 \begin{enumerate}
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94 \item Do the good results previously obtained with deep architectures on the
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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95 MNIST digits generalize to the setting of a much larger and richer (but similar)
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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96 dataset, the NIST special database 19, with 62 classes and around 800k examples?
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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97 \item To what extent does the perturbation of input images (e.g. adding
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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98 noise, affine transformations, background images) make the resulting
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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99 classifier better not only on similarly perturbed images but also on
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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100 the {\em original clean examples}?
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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101 \item Do deep architectures benefit more from such {\em out-of-distribution}
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102 examples, i.e. do they benefit more from the self-taught learning~\citep{RainaR2007} framework?
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103 \item Similarly, does the feature learning step in deep learning algorithms benefit more
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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104 training with similar but different classes (i.e. a multi-task learning scenario) than
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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105 a corresponding shallow and purely supervised architecture?
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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106 \end{enumerate}
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107 The experimental results presented here provide positive evidence towards all of these questions.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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108
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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109 \section{Perturbation and Transformation of Character Images}
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110
467
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111 This section describes the different transformations we used to stochastically
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112 transform source images in order to obtain data. More details can
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113 be found in this technical report~\citep{ift6266-tr-anonymous}.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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114 The code for these transformations (mostly python) is available at
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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115 {\tt http://anonymous.url.net}. All the modules in the pipeline share
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Yoshua Bengio <bengioy@iro.umontreal.ca>
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116 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
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117 amount of deformation or noise introduced.
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118
467
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119 There are two main parts in the pipeline. The first one,
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120 from slant to pinch below, performs transformations. The second
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121 part, from blur to contrast, adds different kinds of noise.
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122
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123 {\large\bf Transformations}\\
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124 {\bf Slant}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
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125 We mimic slant by shifting each row of the image
Yoshua Bengio <bengioy@iro.umontreal.ca>
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126 proportionnaly to its height: $shift = round(slant \times height)$.
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127 The $slant$ coefficient can be negative or positive with equal probability
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128 and its value is randomly sampled according to the complexity level:
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129 e $slant \sim U[0,complexity]$, so the
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130 maximum displacement for the lowest or highest pixel line is of
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131 $round(complexity \times 32)$.\\
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132 {\bf Thickness}\\
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133 Morpholigical operators of dilation and erosion~\citep{Haralick87,Serra82}
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
134 are applied. The neighborhood of each pixel is multiplied
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
135 element-wise with a {\em structuring element} matrix.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
136 The pixel value is replaced by the maximum or the minimum of the resulting
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
137 matrix, respectively for dilation or erosion. Ten different structural elements with
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
138 increasing dimensions (largest is $5\times5$) were used. For each image,
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
139 randomly sample the operator type (dilation or erosion) with equal probability and one structural
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
140 element from a subset of the $n$ smallest structuring elements where $n$ is
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
141 $round(10 \times complexity)$ for dilation and $round(6 \times complexity)$
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
142 for erosion. A neutral element is always present in the set, and if it is
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
143 chosen no transformation is applied. Erosion allows only the six
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
144 smallest structural elements because when the character is too thin it may
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
145 be completely erased.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
146 {\bf Affine Transformations}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
147 A $2 \times 3$ affine transform matrix (with
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
148 6 parameters $(a,b,c,d,e,f)$) is sampled according to the $complexity$ level.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
149 Each pixel $(x,y)$ of the output image takes the value of the pixel
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
150 nearest to $(ax+by+c,dx+ey+f)$ in the input image. This
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
151 produces scaling, translation, rotation and shearing.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
152 The marginal distributions of $(a,b,c,d,e,f)$ have been tuned by hand to
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
153 forbid important rotations (not to confuse classes) but to give good
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
154 variability of the transformation: $a$ and $d$ $\sim U[1-3 \times
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
155 complexity,1+3 \times complexity]$, $b$ and $e$ $\sim[-3 \times complexity,3
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
156 \times complexity]$ and $c$ and $f$ $\sim U[-4 \times complexity, 4 \times
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
157 complexity]$.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
158 {\bf Local Elastic Deformations}\\
469
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
159 This filter induces a "wiggly" effect in the image, following~\citet{SimardSP03},
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
160 which provides more details.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
161 Two "displacements" fields are generated and applied, for horizontal
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
162 and vertical displacements of pixels.
464
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
163 To generate a pixel in either field, first a value between -1 and 1 is
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
164 chosen from a uniform distribution. Then all the pixels, in both fields, are
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
165 multiplied by a constant $\alpha$ which controls the intensity of the
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
166 displacements (larger $\alpha$ translates into larger wiggles).
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
167 Each field is convoluted with a Gaussian 2D kernel of
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
168 standard deviation $\sigma$. Visually, this results in a blur.
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
169 $\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
170 \sqrt[3]{complexity}$.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
171 {\bf Pinch}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
172 This GIMP filter is named "Whirl and
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
173 pinch", but whirl was set to 0. A pinch is ``similar to projecting the image onto an elastic
469
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
174 surface and pressing or pulling on the center of the surface''~\citep{GIMP-manual}.
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
175 For a square input image, think of drawing a circle of
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
176 radius $r$ around a center point $C$. Any point (pixel) $P$ belonging to
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
177 that disk (region inside circle) will have its value recalculated by taking
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
178 the value of another "source" pixel in the original image. The position of
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
179 that source pixel is found on the line thats goes through $C$ and $P$, but
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
180 at some other distance $d_2$. Define $d_1$ to be the distance between $P$
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
181 and $C$. $d_2$ is given by $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
182 d_1$, where $pinch$ is a parameter to the filter.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
183 The actual value is given by bilinear interpolation considering the pixels
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
184 around the (non-integer) source position thus found.
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
185 Here $pinch \sim U[-complexity, 0.7 \times complexity]$.\\
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
186
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
187 {\large\bf Injecting Noise}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
188 {\bf Motion Blur}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
189 This GIMP filter is a ``linear motion blur'' in GIMP
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
190 terminology, with two parameters, $length$ and $angle$. The value of
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
191 a pixel in the final image is the approximately mean value of the $length$ first pixels
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
192 found by moving in the $angle$ direction.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
193 Here $angle \sim U[0,360]$ degrees, and $length \sim {\rm Normal}(0,(3 \times complexity)^2)$.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
194 {\bf Occlusion}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
195 This filter selects a random rectangle from an {\em occluder} character
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
196 images and places it over the original {\em occluded} character
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
197 image. Pixels are combined by taking the max(occluder,occluded),
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
198 closer to black. The corners of the occluder The rectangle corners
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
199 are sampled so that larger complexity gives larger rectangles.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
200 The destination position in the occluded image are also sampled
469
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
201 according to a normal distribution (see more details in~\citet{ift6266-tr-anonymous}).
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
202 It has has a probability of not being applied at all of 60\%.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
203 {\bf Pixel Permutation}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
204 This filter permutes neighbouring pixels. It selects first
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
205 $\frac{complexity}{3}$ pixels randomly in the image. Each of them are then
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
206 sequentially exchanged to one other pixel in its $V4$ neighbourhood. Number
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
207 of exchanges to the left, right, top, bottom are equal or does not differ
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
208 from more than 1 if the number of selected pixels is not a multiple of 4.
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
209 It has has a probability of not being applied at all of 80\%.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
210 {\bf Gaussian Noise}\\
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
211 This filter simply adds, to each pixel of the image independently, a
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
212 noise $\sim Normal(0(\frac{complexity}{10})^2)$.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
213 It has has a probability of not being applied at all of 70\%.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
214 {\bf Background Images}\\
469
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
215 Following~\citet{Larochelle-jmlr-2009}, this transformation adds a random
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
216 background behind the letter. The background is chosen by first selecting,
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
217 at random, an image from a set of images. Then a 32$\times$32 subregion
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
218 of that image is chosen as the background image (by sampling position
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
219 uniformly while making sure not to cross image borders).
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
220 To combine the original letter image and the background image, contrast
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
221 adjustments are made. We first get the maximal values (i.e. maximal
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
222 intensity) for both the original image and the background image, $maximage$
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
223 and $maxbg$. We also have a parameter $contrast \sim U[complexity, 1]$.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
224 Each background pixel value is multiplied by $\frac{max(maximage -
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
225 contrast, 0)}{maxbg}$ (higher contrast yield darker
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
226 background). The output image pixels are max(background,original).\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
227 {\bf Salt and Pepper Noise}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
228 This filter adds noise $\sim U[0,1]$ to random subsets of pixels.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
229 The number of selected pixels is $0.2 \times complexity$.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
230 This filter has a probability of not being applied at all of 75\%.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
231 {\bf Spatially Gaussian Noise}\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
232 Different regions of the image are spatially smoothed.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
233 The image is convolved with a symmetric Gaussian kernel of
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
234 size and variance choosen uniformly in the ranges $[12,12 + 20 \times
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
235 complexity]$ and $[2,2 + 6 \times complexity]$. The result is normalized
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
236 between $0$ and $1$. We also create a symmetric averaging window, of the
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
237 kernel size, with maximum value at the center. For each image we sample
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
238 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
239 averaging centers between the original image and the filtered one. We
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
240 initialize to zero a mask matrix of the image size. For each selected pixel
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
241 we add to the mask the averaging window centered to it. The final image is
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
242 computed from the following element-wise operation: $\frac{image + filtered
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
243 image \times mask}{mask+1}$.
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
244 This filter has a probability of not being applied at all of 75\%.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
245 {\bf Scratches}\\
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
246 The scratches module places line-like white patches on the image. The
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
247 lines are heavily transformed images of the digit "1" (one), chosen
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
248 at random among five thousands such 1 images. The 1 image is
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
249 randomly cropped and rotated by an angle $\sim Normal(0,(100 \times
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
250 complexity)^2$, using bicubic interpolation,
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
251 Two passes of a greyscale morphological erosion filter
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
252 are applied, reducing the width of the line
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
253 by an amount controlled by $complexity$.
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
254 This filter is only applied only 15\% of the time. When it is applied, 50\%
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
255 of the time, only one patch image is generated and applied. In 30\% of
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
256 cases, two patches are generated, and otherwise three patches are
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
257 generated. The patch is applied by taking the maximal value on any given
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
258 patch or the original image, for each of the 32x32 pixel locations.\\
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
259 {\bf Color and Contrast Changes}\\
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
260 This filter changes the constrast and may invert the image polarity (white
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
261 on black to black on white). The contrast $C$ is defined here as the
467
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
262 difference between the maximum and the minimum pixel value of the image.
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
263 Contrast $\sim U[1-0.85 \times complexity,1]$ (so constrast $\geq 0.15$).
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 466
diff changeset
264 The image is normalized into $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
265 polarity is inverted with $0.5$ probability.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
266
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
267
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
268 \begin{figure}[h]
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
269 \resizebox{.99\textwidth}{!}{\includegraphics{images/example_t.png}}\\
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
270 \caption{Illustration of the pipeline of stochastic
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
271 transformations applied to the image of a lower-case t
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
272 (the upper left image). Each image in the pipeline (going from
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
273 left to right, first top line, then bottom line) shows the result
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
274 of applying one of the modules in the pipeline. The last image
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
275 (bottom right) is used as training example.}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
276 \label{fig:pipeline}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
277 \end{figure}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
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278
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
279
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
280 \section{Experimental Setup}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
281
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
282 \subsection{Training Datasets}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
283
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
284 \subsubsection{Data Sources}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
285
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
286 \begin{itemize}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
287 \item {\bf NIST}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
288 The NIST Special Database 19 (NIST19) is a very widely used dataset for training and testing OCR systems.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
289 The dataset is composed with 8????? digits and characters (upper and lower cases), with hand checked classifications,
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
290 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
291 corresponding to "0"-"9","A"-"Z" and "a"-"z". The dataset contains 8 series of different complexity.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
292 The fourth series, $hsf_4$, experimentally recognized to be the most difficult one for classification task is recommended
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
293 by NIST as testing set and is used in our work for that purpose. It contains 82600 examples,
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
294 while the training and validation sets (which have the same distribution) contain XXXXX and
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
295 XXXXX examples respectively.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
296 The performances reported by previous work on that dataset mostly use only the digits.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
297 Here we use all the classes both in the training and testing phase. This is especially
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
298 useful to estimate the effect of a multi-task setting.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
299 Note that the distribution of the classes in the NIST training and test sets differs
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
300 substantially, with relatively many more digits in the test set, and uniform distribution
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
301 of letters in the test set, not in the training set (more like the natural distribution
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
302 of letters in text).
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
303
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
304 \item {\bf Fonts} TODO!!!
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
305
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
306 \item {\bf Captchas}
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
307 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
308 generating characters of the same format as the NIST dataset. The core of this data source is composed with a random character
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
309 generator and various kinds of tranformations similar to those described in the previous sections.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
310 In order to increase the variability of the data generated, different fonts are used for generating the characters.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
311 Transformations (slant, distorsions, rotation, translation) are applied to each randomly generated character with a complexity
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
312 depending on the value of the complexity parameter provided by the user of the data source. Two levels of complexity are
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
313 allowed and can be controlled via an easy to use facade class.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
314 \item {\bf OCR data}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
315 \end{itemize}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
316
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
317 \subsubsection{Data Sets}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
318 \begin{itemize}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
319 \item {\bf NIST} This is the raw NIST special database 19.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
320 \item {\bf P07}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
321 The dataset P07 is sampled with our transformation pipeline with a complexity parameter of $0.7$.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
322 For each new exemple to generate, we choose one source with the following probability: $0.1$ for the fonts,
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
323 $0.25$ for the captchas, $0.25$ for OCR data and $0.4$ for NIST. We apply all the transformations in their order
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
324 and for each of them we sample uniformly a complexity in the range $[0,0.7]$.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
325 \item {\bf NISTP} NISTP is equivalent to P07 (complexity parameter of $0.7$ with the same sources proportion)
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
326 except that we only apply
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
327 transformations from slant to pinch. Therefore, the character is
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
328 transformed but no additionnal noise is added to the image, giving images
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
329 closer to the NIST dataset.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
330 \end{itemize}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
331
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
332 \subsection{Models and their Hyperparameters}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
333
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
334 \subsubsection{Multi-Layer Perceptrons (MLP)}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
335
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
336 An MLP is a family of functions that are described by stacking layers of of a function similar to
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
337 $$g(x) = \tanh(b+Wx)$$
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
338 The input, $x$, is a $d$-dimension vector.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
339 The output, $g(x)$, is a $m$-dimension vector.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
340 The parameter $W$ is a $m\times d$ matrix and is called the weight matrix.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
341 The parameter $b$ is a $m$-vector and is called the bias vector.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
342 The non-linearity (here $\tanh$) is applied element-wise to the output vector.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
343 Usually the input is referred to a input layer and similarly for the output.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
344 You can of course chain several such functions to obtain a more complex one.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
345 Here is a common example
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
346 $$f(x) = c + V\tanh(b+Wx)$$
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
347 In this case the intermediate layer corresponding to $\tanh(b+Wx)$ is called a hidden layer.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
348 Here the output layer does not have the same non-linearity as the hidden layer.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
349 This is a common case where some specialized non-linearity is applied to the output layer only depending on the task at hand.
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
350
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
351 If you put 3 or more hidden layers in such a network you obtain what is called a deep MLP.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
352 The parameters to adapt are the weight matrix and the bias vector for each layer.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
353
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
354 \subsubsection{Stacked Denoising Auto-Encoders (SDAE)}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
355 \label{SdA}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
356
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
357 Auto-encoders are essentially a way to initialize the weights of the network to enable better generalization.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
358 This is essentially unsupervised training where the layer is made to reconstruct its input through and encoding and decoding phase.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
359 Denoising auto-encoders are a variant where the input is corrupted with random noise but the target is the uncorrupted input.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
360 The principle behind these initialization methods is that the network will learn the inherent relation between portions of the data and be able to represent them thus helping with whatever task we want to perform.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
361
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
362 An auto-encoder unit is formed of two MLP layers with the bottom one called the encoding layer and the top one the decoding layer.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
363 Usually the top and bottom weight matrices are the transpose of each other and are fixed this way.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
364 The network is trained as such and, when sufficiently trained, the MLP layer is initialized with the parameters of the encoding layer.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
365 The other parameters are discarded.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
366
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
367 The stacked version is an adaptation to deep MLPs where you initialize each layer with a denoising auto-encoder starting from the bottom.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
368 During the initialization, which is usually called pre-training, the bottom layer is treated as if it were an isolated auto-encoder.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
369 The second and following layers receive the same treatment except that they take as input the encoded version of the data that has gone through the layers before it.
469
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
370 For additional details see \citet{vincent:icml08}.
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
371
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
372 \section{Experimental Results}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
373
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
374 \subsection{SDA vs MLP vs Humans}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
375
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
376 We compare here the best MLP (according to validation set error) that we found against
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
377 the best SDA (again according to validation set error), along with a precise estimate
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
378 of human performance obtained via Amazon's Mechanical Turk (AMT)
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
379 service\footnote{http://mturk.com}. AMT users are paid small amounts
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
380 of money to perform tasks for which human intelligence is required.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
381 Mechanical Turk has been used extensively in natural language
469
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
382 processing \citep{SnowEtAl2008} and vision
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
383 \citep{SorokinAndForsyth2008,whitehill09}. AMT users where presented
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
384 with 10 character images and asked to type 10 corresponding ascii
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
385 characters. Hence they were forced to make a hard choice among the
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
386 62 character classes. Three users classified each image, allowing
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
387 to estimate inter-human variability (shown as +/- in parenthesis below).
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
388
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
389 \begin{table}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
390 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits +
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
391 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
392 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
393 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07)
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
394 and using a validation set to select hyper-parameters and other training choices.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
395 \{SDA,MLP\}0 are trained on NIST,
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
396 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
397 The human error rate on digits is a lower bound because it does not count digits that were
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
398 recognized as letters. For comparison, the results found in the literature
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
399 on NIST digits classification using the same test set are included.}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
400 \label{tab:sda-vs-mlp-vs-humans}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
401 \begin{center}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
402 \begin{tabular}{|l|r|r|r|r|} \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
403 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
404 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $>1.1\%$ \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
405 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
406 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
407 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
408 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
409 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
410 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline
469
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
411 \citep{Granger+al-2007} & & & & 4.95\% $\pm$.18\% \\ \hline
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
412 \citep{Cortes+al-2000} & & & & 3.71\% $\pm$.16\% \\ \hline
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
413 \citep{Oliveira+al-2002} & & & & 2.4\% $\pm$.13\% \\ \hline
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
414 \citep{Migram+al-2005} & & & & 2.1\% $\pm$.12\% \\ \hline
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
415 \end{tabular}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
416 \end{center}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
417 \end{table}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
418
475
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parents: 469
diff changeset
419 \begin{figure}[h]
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
420 \resizebox{.99\textwidth}{!}{\includegraphics{images/error_rates_charts.pdf}}\\
ead3085c1c66 Added charts to nips2010_submission.tex
fsavard
parents: 469
diff changeset
421 \caption{Charts corresponding to table \ref{tab:sda-vs-mlp-vs-humans}. Left: overall results; error bars indicate a 95\% confidence interval. Right: error rates on NIST test digits only, with results from litterature. }
ead3085c1c66 Added charts to nips2010_submission.tex
fsavard
parents: 469
diff changeset
422 \label{fig:error-rates-charts}
ead3085c1c66 Added charts to nips2010_submission.tex
fsavard
parents: 469
diff changeset
423 \end{figure}
ead3085c1c66 Added charts to nips2010_submission.tex
fsavard
parents: 469
diff changeset
424
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
425 \subsection{Perturbed Training Data More Helpful for SDAE}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
426
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
427 \begin{table}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
428 \caption{Relative change in error rates due to the use of perturbed training data,
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
429 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
430 A positive value indicates that training on the perturbed data helped for the
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
431 given test set (the first 3 columns on the 62-class tasks and the last one is
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
432 on the clean 10-class digits). Clearly, the deep learning models did benefit more
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
433 from perturbed training data, even when testing on clean data, whereas the MLP
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
434 trained on perturbed data performed worse on the clean digits and about the same
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
435 on the clean characters. }
469
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
436 \label{tab:perturbation-effect}
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
437 \begin{center}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
438 \begin{tabular}{|l|r|r|r|r|} \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
439 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
440 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
441 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
442 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
443 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
444 \end{tabular}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
445 \end{center}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
446 \end{table}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
447
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
448
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
449 \subsection{Multi-Task Learning Effects}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
450
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
451 As previously seen, the SDA is better able to benefit from the
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
452 transformations applied to the data than the MLP. In this experiment we
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
453 define three tasks: recognizing digits (knowing that the input is a digit),
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
454 recognizing upper case characters (knowing that the input is one), and
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
455 recognizing lower case characters (knowing that the input is one). We
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
456 consider the digit classification task as the target task and we want to
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
457 evaluate whether training with the other tasks can help or hurt, and
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
458 whether the effect is different for MLPs versus SDAs. The goal is to find
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
459 out if deep learning can benefit more (or less) from multiple related tasks
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
460 (i.e. the multi-task setting) compared to a corresponding purely supervised
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
461 shallow learner.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
462
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
463 We use a single hidden layer MLP with 1000 hidden units, and a SDA
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
464 with 3 hidden layers (1000 hidden units per layer), pre-trained and
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
465 fine-tuned on NIST.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
466
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
467 Our results show that the MLP benefits marginally from the multi-task setting
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
468 in the case of digits (5\% relative improvement) but is actually hurt in the case
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
469 of characters (respectively 3\% and 4\% worse for lower and upper class characters).
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
470 On the other hand the SDA benefitted from the multi-task setting, with relative
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
471 error rate improvements of 27\%, 15\% and 13\% respectively for digits,
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
472 lower and upper case characters, as shown in Table~\ref{tab:multi-task}.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
473
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
474 \begin{table}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
475 \caption{Test error rates and relative change in error rates due to the use of
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
476 a multi-task setting, i.e., training on each task in isolation vs training
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
477 for all three tasks together, for MLPs vs SDAs. The SDA benefits much
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
478 more from the multi-task setting. All experiments on only on the
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
479 unperturbed NIST data, using validation error for model selection.
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
480 Relative improvement is 1 - single-task error / multi-task error.}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
481 \label{tab:multi-task}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
482 \begin{center}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
483 \begin{tabular}{|l|r|r|r|} \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
484 & single-task & multi-task & relative \\
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
485 & setting & setting & improvement \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
486 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
487 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
488 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
489 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
490 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
491 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
492 \end{tabular}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
493 \end{center}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
494 \end{table}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
495
475
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parents: 469
diff changeset
496
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
497 \begin{figure}[h]
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
498 \resizebox{.99\textwidth}{!}{\includegraphics{images/improvements_charts.pdf}}\\
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
499 \caption{Charts corresponding to tables \ref{tab:perturbation-effect} (left) and \ref{tab:multi-task} (right).}
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
500 \label{fig:improvements-charts}
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
501 \end{figure}
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
502
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
503
ead3085c1c66 Added charts to nips2010_submission.tex
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parents: 469
diff changeset
504
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
505 \section{Conclusions}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
506
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
507 \bibliography{strings,ml,aigaion,specials}
469
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
508 %\bibliographystyle{plainnat}
d02d288257bf redone bib style
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents: 467
diff changeset
509 \bibliographystyle{unsrtnat}
464
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
510 %\bibliographystyle{apalike}
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
511
24f4a8b53fcc nips2010_submission.tex
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff changeset
512 \end{document}