A convergent Deep Learning algorithm for approximation of polynomials
We start from the contractive functional equation proposed in [4], where it was shown that the polynomial solution of functional equation can be used to initialize a Neural Network structure, with a controlled accuracy. We propose a novel algorithm, where the functional equation is solved with a con...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-09-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.462/ |
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| author | Després, Bruno |
| author_facet | Després, Bruno |
| author_sort | Després, Bruno |
| collection | DOAJ |
| description | We start from the contractive functional equation proposed in [4], where it was shown that the polynomial solution of functional equation can be used to initialize a Neural Network structure, with a controlled accuracy. We propose a novel algorithm, where the functional equation is solved with a converging iterative algorithm which can be realized as a Machine Learning training method iteratively with respect to the number of layers. The proof of convergence is performed with respect to the $L^\infty $ norm. Numerical tests illustrate the theory and show that stochastic gradient descent methods can be used with good accuracy for this problem. |
| format | Article |
| id | doaj-art-294b157f688945229ca040a60f4ad51b |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-294b157f688945229ca040a60f4ad51b2025-02-07T11:09:17ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-09-01361G61029104010.5802/crmath.46210.5802/crmath.462A convergent Deep Learning algorithm for approximation of polynomialsDesprés, Bruno0Laboratoire Jacques-Louis Lions, Sorbonne Université, 4 place Jussieu, 75005 Paris, FranceWe start from the contractive functional equation proposed in [4], where it was shown that the polynomial solution of functional equation can be used to initialize a Neural Network structure, with a controlled accuracy. We propose a novel algorithm, where the functional equation is solved with a converging iterative algorithm which can be realized as a Machine Learning training method iteratively with respect to the number of layers. The proof of convergence is performed with respect to the $L^\infty $ norm. Numerical tests illustrate the theory and show that stochastic gradient descent methods can be used with good accuracy for this problem.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.462/ |
| spellingShingle | Després, Bruno A convergent Deep Learning algorithm for approximation of polynomials Comptes Rendus. Mathématique |
| title | A convergent Deep Learning algorithm for approximation of polynomials |
| title_full | A convergent Deep Learning algorithm for approximation of polynomials |
| title_fullStr | A convergent Deep Learning algorithm for approximation of polynomials |
| title_full_unstemmed | A convergent Deep Learning algorithm for approximation of polynomials |
| title_short | A convergent Deep Learning algorithm for approximation of polynomials |
| title_sort | convergent deep learning algorithm for approximation of polynomials |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.462/ |
| work_keys_str_mv | AT despresbruno aconvergentdeeplearningalgorithmforapproximationofpolynomials AT despresbruno convergentdeeplearningalgorithmforapproximationofpolynomials |