X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/hdrcouchot.git/blobdiff_plain/b7ce5574dead7f7c53fe6362ac1655d8d54fcd0c..b78b2e8df79b2ff2b99b448379585acd37c823d4:/chaosANN.tex diff --git a/chaosANN.tex b/chaosANN.tex index 22d5258..239251e 100644 --- a/chaosANN.tex +++ b/chaosANN.tex @@ -158,7 +158,7 @@ de vérifier si le graphe d'itérations $\textsc{giu}(f)$ 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 @@ -176,7 +176,7 @@ de de fonction à quatre éléments. \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*} @@ -268,290 +268,162 @@ ont montré qu'un MLP pouvait apprendre la dynamique du circuit de Chua. 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}