$\textsc{giu}(f)$ est fortement connexe.
-\section{Un réseau de neurones peut-il approximer un
+\section{Un réseau de neurones peut-il approximer
des itération unaires chaotiques?}
Cette section s'intéresse à étudier le comportement d'un réseau de neurones
\subsection{Construction du réseau}
\label{section:translation}
-On considère par exemple les deux fonctions $f$ and $g$ de0 $\Bool^4$
+On considère par exemple les deux fonctions $f$ and $g$ de $\Bool^4$
dans $\Bool^4$ définies par:
\begin{eqnarray*}
Ce réseau avec rétropropagation est composé de deux couches
et entrainé à l'aide d'une propagation arrière Bayesienne.
-In these experiments we consider MLPs having one hidden layer of
-sigmoidal neurons and output neurons with a linear activation
-function. They are trained using the Limited-memory
-Broyden-Fletcher-Goldfarb-Shanno quasi-newton algorithm in combination
-with the Wolfe linear search. The training process is performed until
-a maximum number of epochs is reached. To prevent overfitting and to
-estimate the generalization performance we use holdout validation by
-splitting the data set into learning, validation, and test subsets.
-These subsets are obtained through random selection such that their
-respective size represents 65\%, 10\%, and 25\% of the whole data set.
-
-Several neural networks are trained for both iterations coding
-schemes. In both cases iterations have the following layout:
-configurations of four components and strategies with at most three
-terms. Thus, for the first coding scheme a data set pair is composed
-of 6~inputs and 5~outputs, while for the second one it is respectively
-3~inputs and 2~outputs. As noticed at the end of the previous section,
-this leads to data sets that consist of 2304~pairs. The networks
-differ in the size of the hidden layer and the maximum number of
-training epochs. We remember that to evaluate the ability of neural
-networks to predict a chaotic behavior for each coding scheme, the
-trainings of two data sets, one of them describing chaotic iterations,
-are compared.
-
-Thereafter we give, for the different learning setups and data sets,
-the mean prediction success rate obtained for each output. Such a rate
-represents the percentage of input-output pairs belonging to the test
-subset for which the corresponding output value was correctly
-predicted. These values are computed considering 10~trainings with
-random subsets construction, weights and biases initialization.
-Firstly, neural networks having 10 and 25~hidden neurons are trained,
-with a maximum number of epochs that takes its value in
-$\{125,250,500\}$ (see Tables~\ref{tab1} and \ref{tab2}). Secondly,
-we refine the second coding scheme by splitting the output vector such
-that each output is learned by a specific neural network
-(Table~\ref{tab3}). In this last case, we increase the size of the
-hidden layer up to 40~neurons and we consider larger number of epochs.
+Le choix de l'achitecture du réseau ainsi que de la méthode d'apprentissage
+ont été détaillé dans~\cite{bcgs12:ij}.
+En pratique, nous avons considéré des configurations de
+quatre éléments booléens
+et une stratégie fixe de longueur 3.
+Pour le premier codage, nous avons ainsi 6~entrées et 5~sorties
+tandis que pour le second, uniquement 3 entrées et 2 sorties.
+Cela engendre ainsi 2304~combinaisons possibles comme détaillé à la
+section précédente.
+
+
\begin{table}[htbp!]
-\caption{Prediction success rates for configurations expressed as boolean vectors.}
-\label{tab1}
\centering {\small
\begin{tabular}{|c|c||c|c|c|}
\hline
-\multicolumn{5}{|c|}{Networks topology: 6~inputs, 5~outputs, and one hidden layer} \\
+\multicolumn{5}{|c|}{Topologie du réseau: 6~entrées, 5~sorties, 1~couche cachée} \\
\hline
\hline
-\multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{10 neurons} \\
+\multicolumn{2}{|c||}{Neurones cachés} & \multicolumn{3}{c|}{10 neurones} \\
\cline{3-5}
\multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Chaotic}}&Output~(1) & 90.92\% & 91.75\% & 91.82\% \\
-& Output~(2) & 69.32\% & 78.46\% & 82.15\% \\
-& Output~(3) & 68.47\% & 78.49\% & 82.22\% \\
-& Output~(4) & 91.53\% & 92.37\% & 93.4\% \\
+\multirow{6}{*}{\rotatebox{90}{Chaotique $g$ }}&Entrée~(1) & 90.92\% & 91.75\% & 91.82\% \\
+& Entrée~(2) & 69.32\% & 78.46\% & 82.15\% \\
+& Entrée~(3) & 68.47\% & 78.49\% & 82.22\% \\
+& Entrée~(4) & 91.53\% & 92.37\% & 93.4\% \\
& Config. & 36.10\% & 51.35\% & 56.85\% \\
-& Strategy~(5) & 1.91\% & 3.38\% & 2.43\% \\
+& Stratégie~(5) & 1.91\% & 3.38\% & 2.43\% \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Non-chaotic}}&Output~(1) & 97.64\% & 98.10\% & 98.20\% \\
-& Output~(2) & 95.15\% & 95.39\% & 95.46\% \\
-& Output~(3) & 100\% & 100\% & 100\% \\
-& Output~(4) & 97.47\% & 97.90\% & 97.99\% \\
+\multirow{6}{*}{\rotatebox{90}{Non-chaotic $f$}}&Entrée~(1) & 97.64\% & 98.10\% & 98.20\% \\
+& Entrée~(2) & 95.15\% & 95.39\% & 95.46\% \\
+& Entrée~(3) & 100\% & 100\% & 100\% \\
+& Entrée~(4) & 97.47\% & 97.90\% & 97.99\% \\
& Config. & 90.52\% & 91.59\% & 91.73\% \\
-& Strategy~(5) & 3.41\% & 3.40\% & 3.47\% \\
+& Stratégie~(5) & 3.41\% & 3.40\% & 3.47\% \\
\hline
\hline
-\multicolumn{2}{|c||}{Hidden neurons} & \multicolumn{3}{c|}{25 neurons} \\ %& \multicolumn{3}{|c|}{40 neurons} \\
-\cline{3-5}
+\multicolumn{2}{|c||}{Neurones cachés} & \multicolumn{3}{c|}{25 neurones} \\
+\cline{3-5} \\%& \multicolumn{3}{|c|}{40 neurons} \\
\multicolumn{2}{|c||}{Epochs} & 125 & 250 & 500 \\ %& 125 & 250 & 500 \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Chaotic}}&Output~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
-& Output~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
-& Output~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
-& Output~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
+\multirow{6}{*}{\rotatebox{90}{Chaotique $g$}}&Entrée~(1) & 91.65\% & 92.69\% & 93.93\% \\ %& 91.94\% & 92.89\% & 94.00\% \\
+& Entrée~(2) & 72.06\% & 88.46\% & 90.5\% \\ %& 74.97\% & 89.83\% & 91.14\% \\
+& Entrée~(3) & 79.19\% & 89.83\% & 91.59\% \\ %& 76.69\% & 89.58\% & 91.84\% \\
+& Entrée~(4) & 91.61\% & 92.34\% & 93.47\% \\% & 82.77\% & 92.93\% & 93.48\% \\
& Config. & 48.82\% & 67.80\% & 70.97\% \\%& 49.46\% & 68.94\% & 71.11\% \\
-& Strategy~(5) & 2.62\% & 3.43\% & 3.78\% \\% & 3.10\% & 3.10\% & 3.03\% \\
+& Stratégie~(5) & 2.62\% & 3.43\% & 3.78\% \\% & 3.10\% & 3.10\% & 3.03\% \\
\hline
-\multirow{6}{*}{\rotatebox{90}{Non-chaotic}}&Output~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
-& Output~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
-& Output~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
-& Output~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
+\multirow{6}{*}{\rotatebox{90}{Non-chaotique $f$}}&Entrée~(1) & 97.87\% & 97.99\% & 98.03\% \\ %& 98.16\% \\
+& Entrée~(2) & 95.46\% & 95.84\% & 96.75\% \\ % & 97.4\% \\
+& Entrée~(3) & 100\% & 100\% & 100\% \\%& 100\% \\
+& Entrée~(4) & 97.77\% & 97.82\% & 98.06\% \\%& 98.31\% \\
& Config. & 91.36\% & 91.99\% & 93.03\% \\%& 93.98\% \\
-& Strategy~(5) & 3.37\% & 3.44\% & 3.29\% \\%& 3.23\% \\
+& Stratégie~(5) & 3.37\% & 3.44\% & 3.29\% \\%& 3.23\% \\
\hline
\end{tabular}
}
+\caption{Taux de prédiction lorsque les configurations sont exprimées comme un vecteur booléen.}
+\label{tab1}
\end{table}
+Le tableau~\ref{tab1} synthétise les résultats obtenus avec le premier
+codage. Sans surprise, la précision de la prédiction croit
+avec l'Epoch et le nombre de neurones sur la couche cachée.
+Dans tous les cas, les résultats sont plus précis dans le cas non
+chaotique que dans l'autre. Enfin, le réseau ne parvient jamais à
+apprendre le comportement de la stratégie.
-Table~\ref{tab1} presents the rates obtained for the first coding
-scheme. For the chaotic data, it can be seen that as expected
-configuration prediction becomes better when the number of hidden
-neurons and maximum epochs increases: an improvement by a factor two
-is observed (from 36.10\% for 10~neurons and 125~epochs to 70.97\% for
-25~neurons and 500~epochs). We also notice that the learning of
-outputs~(2) and~(3) is more difficult. Conversely, for the
-non-chaotic case the simplest training setup is enough to predict
-configurations. For all these feedforward network topologies and all
-outputs the obtained results for the non-chaotic case outperform the
-chaotic ones. Finally, the rates for the strategies show that the
-different feedforward networks are unable to learn them.
-
-For the second coding scheme (\textit{i.e.}, with Gray Codes)
-Table~\ref{tab2} shows that any network learns about five times more
-non-chaotic configurations than chaotic ones. As in the previous
-scheme, the strategies cannot be predicted.
-Figures~\ref{Fig:chaotic_predictions} and
-\ref{Fig:non-chaotic_predictions} present the predictions given by two
-feedforward multilayer perceptrons that were respectively trained to
-learn chaotic and non-chaotic data, using the second coding scheme.
-Each figure shows for each sample of the test subset (577~samples,
-representing 25\% of the 2304~samples) the configuration that should
-have been predicted and the one given by the multilayer perceptron. It
-can be seen that for the chaotic data the predictions are far away
-from the expected configurations. Obviously, the better predictions
-for the non-chaotic data reflect their regularity.
-
-Let us now compare the two coding schemes. Firstly, the second scheme
-disturbs the learning process. In fact in this scheme the
-configuration is always expressed as a natural number, whereas in the
-first one the number of inputs follows the increase of the Boolean
-vectors coding configurations. In this latter case, the coding gives a
-finer information on configuration evolution.
\begin{table}[b]
-\caption{Prediction success rates for configurations expressed with Gray code}
-\label{tab2}
\centering
\begin{tabular}{|c|c||c|c|c|}
\hline
-\multicolumn{5}{|c|}{Networks topology: 3~inputs, 2~outputs, and one hidden layer} \\
+\multicolumn{5}{|c|}{Topologie du réseau: 3~entrées, 2~sorties, 1~couche cachée} \\
\hline
\hline
-& Hidden neurons & \multicolumn{3}{c|}{10 neurons} \\
+& Neurones cachés & \multicolumn{3}{c|}{10 neurones} \\
\cline{2-5}
& Epochs & 125 & 250 & 500 \\ %& 1000
\hline
-\multirow{2}{*}{Chaotic}& Config.~(1) & 13.29\% & 13.55\% & 13.08\% \\ %& 12.5\%
-& Strategy~(2) & 0.50\% & 0.52\% & 1.32\% \\ %& 1.42\%
+\multirow{2}{*}{Chaotique $g$}& Config.~(1) & 13.29\% & 13.55\% & 13.08\% \\ %& 12.5\%
+& Stratégie~(2) & 0.50\% & 0.52\% & 1.32\% \\ %& 1.42\%
\hline
-\multirow{2}{*}{Non-Chaotic}&Config.~(1) & 77.12\% & 74.00\% & 72.60\% \\ %& 75.81\%
-& Strategy~(2) & 0.42\% & 0.80\% & 1.16\% \\ %& 1.42\%
+\multirow{2}{*}{Non-Chaotique $f$}&Config.~(1) & 77.12\% & 74.00\% & 72.60\% \\ %& 75.81\%
+& Stratégie~(2) & 0.42\% & 0.80\% & 1.16\% \\ %& 1.42\%
\hline
\hline
-& Hidden neurons & \multicolumn{3}{c|}{25 neurons} \\
+& Neurones cachés& \multicolumn{3}{c|}{25 neurones} \\
\cline{2-5}
& Epochs & 125 & 250 & 500 \\ %& 1000
\hline
-\multirow{2}{*}{Chaotic}& Config.~(1) & 12.27\% & 13.15\% & 13.05\% \\ %& 15.44\%
-& Strategy~(2) & 0.71\% & 0.66\% & 0.88\% \\ %& 1.73\%
+\multirow{2}{*}{Chaotique $g$ }& Config.~(1) & 12.27\% & 13.15\% & 13.05\% \\ %& 15.44\%
+& Stratégie~(2) & 0.71\% & 0.66\% & 0.88\% \\ %& 1.73\%
\hline
-\multirow{2}{*}{Non-Chaotic}&Config.~(1) & 73.60\% & 74.70\% & 75.89\% \\ %& 68.32\%
-& Strategy~(2) & 0.64\% & 0.97\% & 1.23\% \\ %& 1.80\%
+\multirow{2}{*}{Non-Chaotique $f$}&Config.~(1) & 73.60\% & 74.70\% & 75.89\% \\ %& 68.32\%
+& Stratégie~(2) & 0.64\% & 0.97\% & 1.23\% \\ %& 1.80\%
\hline
\end{tabular}
+\caption{Taux de prédiction lorsque les configurations sont exprimées
+à l'aide de codes de Gray.}
+\label{tab2}
\end{table}
-\begin{figure}
- \centering
- \includegraphics[scale=0.5]{images/chaotic_trace2}
- \caption {Second coding scheme - Predictions obtained for a chaotic test subset.}
- \label{Fig:chaotic_predictions}
-\end{figure}
-\begin{figure}
- \centering
- \includegraphics[scale=0.5]{images/non-chaotic_trace2}
- \caption{Second coding scheme - Predictions obtained for a non-chaotic test subset.}
- \label{Fig:non-chaotic_predictions}
+
+Les résultats concernant le second codage (\textit{i.e.}, avec les codes
+de Gray) sont synthétisés dans le tableau~\ref{tab2}. On constate
+que le réseau apprend cinq fois mieux les comportement non chaotiques
+que ceux qui le sont. Ceci est est illustré au travers des
+figures~\ref{Fig:chaotic_predictions} et~\ref{Fig:non-chaotic_predictions}.
+De plus, comme dans le codage précédent, les stratégies ne peuvent pas être
+prédites.
+On constate que ce second codage réduit certe le nombre de sorties, mais est
+largement moins performant que le premier.
+On peut expliquer ceci par le fait
+que ce second codage garantit que deux entiers successifs correspondent
+à deux configurations voisines, \textit{ie.e}, qui ne diffèrent que d'un
+élément.
+La réciproque n'est cependant pas établie et deux configurations voisines
+peuvent être traduitent par des entiers très éloignés et ainsi difficils
+àapprendre.
+
+
+\begin{figure}[ht]
+ \begin{center}
+ \subfigure[Fonction chaotique $g$]{
+ \begin{minipage}{0.48\textwidth}
+ \begin{center}
+ \includegraphics[scale=0.37]{images/chaotic_trace2}
+ \end{center}
+ \end{minipage}
+ \label{Fig:chaotic_predictions}
+ }
+ \subfigure[Fonction non-chaotique $f$]{
+ \begin{minipage}{0.48\textwidth}
+ \begin{center}
+ \includegraphics[scale=0.37]{images/non-chaotic_trace2}
+ \end{center}
+ \end{minipage}
+ \label{Fig:non-chaotic_predictions}
+ }
+ \end{center}
+ \caption {Prédiction lorsque les configurations sont exprimées
+à l'aide de codes de Gray.}
\end{figure}
-Unfortunately, in practical applications the number of components is
-usually unknown. Hence, the first coding scheme cannot be used
-systematically. Therefore, we provide a refinement of the second
-scheme: each output is learned by a different ANN. Table~\ref{tab3}
-presents the results for this approach. In any case, whatever the
-considered feedforward network topologies, the maximum epoch number,
-and the kind of iterations, the configuration success rate is slightly
-improved. Moreover, the strategies predictions rates reach almost
-12\%, whereas in Table~\ref{tab2} they never exceed 1.5\%. Despite of
-this improvement, a long term prediction of chaotic iterations still
-appear to be an open issue.
-
-\begin{table}
-\caption{Prediction success rates for split outputs.}
-\label{tab3}
-\centering
-\begin{tabular}{|c||c|c|c|}
-\hline
-\multicolumn{4}{|c|}{Networks topology: 3~inputs, 1~output, and one hidden layer} \\
-\hline
-\hline
-Epochs & 125 & 250 & 500 \\
-\hline
-\hline
-Chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\hline
-10~neurons & 12.39\% & 14.06\% & 14.32\% \\
-25~neurons & 13.00\% & 14.28\% & 14.58\% \\
-40~neurons & 11.58\% & 13.47\% & 14.23\% \\
-\hline
-\hline
-Non chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\cline{2-4}
-%Epochs & 125 & 250 & 500 \\
-\hline
-10~neurons & 76.01\% & 74.04\% & 78.16\% \\
-25~neurons & 76.60\% & 72.13\% & 75.96\% \\
-40~neurons & 76.34\% & 75.63\% & 77.50\% \\
-\hline
-\hline
-Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
-\cline{2-4}
-%Epochs & 125 & 250 & 500 \\
-\hline
-10~neurons & 0.76\% & 0.97\% & 1.21\% \\
-25~neurons & 1.09\% & 0.73\% & 1.79\% \\
-40~neurons & 0.90\% & 1.02\% & 2.15\% \\
-\hline
-\multicolumn{4}{c}{} \\
-\hline
-Epochs & 1000 & 2500 & 5000 \\
-\hline
-\hline
-Chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\hline
-10~neurons & 14.51\% & 15.22\% & 15.22\% \\
-25~neurons & 16.95\% & 17.57\% & 18.46\% \\
-40~neurons & 17.73\% & 20.75\% & 22.62\% \\
-\hline
-\hline
-Non chaotic & \multicolumn{3}{c|}{Output = Configuration} \\
-\cline{2-4}
-%Epochs & 1000 & 2500 & 5000 \\
-\hline
-10~neurons & 78.98\% & 80.02\% & 79.97\% \\
-25~neurons & 79.19\% & 81.59\% & 81.53\% \\
-40~neurons & 79.64\% & 81.37\% & 81.37\% \\
-\hline
-\hline
-Chaotic/non chaotic & \multicolumn{3}{c|}{Output = Strategy} \\
-\cline{2-4}
-%Epochs & 1000 & 2500 & 5000 \\
-\hline
-10~neurons & 3.47\% & 9.98\% & 11.66\% \\
-25~neurons & 3.92\% & 8.63\% & 10.09\% \\
-40~neurons & 3.29\% & 7.19\% & 7.18\% \\
-\hline
-\end{tabular}
-\end{table}
\section{Conclusion}
-
-In this paper, we have established an equivalence between chaotic
-iterations, according to the Devaney's definition of chaos, and a
-class of multilayer perceptron neural networks. Firstly, we have
-described how to build a neural network that can be trained to learn a
-given chaotic map function. Secondly, we found a condition that allow
-to check whether the iterations induced by a function are chaotic or
-not, and thus if a chaotic map is obtained. Thanks to this condition
-our approach is not limited to a particular function. In the dual
-case, we show that checking if a neural network is chaotic consists in
-verifying a property on an associated graph, called the graph of
-iterations. These results are valid for recurrent neural networks
-with a particular architecture. However, we believe that a similar
-work can be done for other neural network architectures. Finally, we
-have discovered at least one family of problems with a reasonable
-size, such that artificial neural networks should not be applied in
-the presence of chaos, due to their inability to learn chaotic
-behaviors in this context. Such a consideration is not reduced to a
-theoretical detail: this family of discrete iterations is concretely
-implemented in a new steganographic method \cite{guyeux10ter}. As
-steganographic detectors embed tools like neural networks to
-distinguish between original and stego contents, our studies tend to
-prove that such detectors might be unable to tackle with chaos-based
-information hiding schemes.
-
-In future work we intend to enlarge the comparison between the
-learning of truly chaotic and non-chaotic behaviors. Other
-computational intelligence tools such as support vector machines will
-be investigated too, to discover which tools are the most relevant
-when facing a truly chaotic phenomenon. A comparison between learning
-rate success and prediction quality will be realized. Concrete
-consequences in biology, physics, and computer science security fields
-will then be stated.
-
+Dans ce chapitre, nous avons établi une simlilitude entre les itérations
+chaotiques et une famille de perceptrons multicouches.
+Nous avons d'abord montré comment construire un réseau de neurones
+ayant un comportement chaotique.
+Nous avons présenté ensuite comment vérifier si un réseau de neurones
+établi était chaotique.
+Nous avons enfin montré en pratique qu'il est difficile pour un
+réseau de neurones d'apprendre le comportement global d'itérations
+chaotiques.
% \appendix{}
% \begin{Def} \label{def2}
