Fixed width treelike neural networks capacity analysis -- generic activations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916118429433856 |
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| author | Stojnic, Mihailo |
| author_facet | Stojnic, Mihailo |
| contents | We consider the capacity of \emph{treelike committee machines} (TCM) neural networks. Relying on Random Duality Theory (RDT), \cite{Stojnictcmspnncaprdt23} recently introduced a generic framework for their capacity analysis. An upgrade based on the so-called \emph{partially lifted} RDT (pl RDT) was then presented in \cite{Stojnictcmspnncapliftedrdt23}. Both lines of work focused on the networks with the most typical, \emph{sign}, activations. Here, on the other hand, we focus on networks with other, more general, types of activations and show that the frameworks of \cite{Stojnictcmspnncaprdt23,Stojnictcmspnncapliftedrdt23} are sufficiently powerful to enable handling of such scenarios as well. In addition to the standard \emph{linear} activations, we uncover that particularly convenient results can be obtained for two very commonly used activations, namely, the \emph{quadratic} and \emph{rectified linear unit (ReLU)} ones. In more concrete terms, for each of these activations, we obtain both the RDT and pl RDT based memory capacities upper bound characterization for \emph{any} given (even) number of the hidden layer neurons, $d$. In the process, we also uncover the following two, rather remarkable, facts: 1) contrary to the common wisdom, both sets of results show that the bounding capacity decreases for large $d$ (the width of the hidden layer) while converging to a constant value; and 2) the maximum bounding capacity is achieved for the networks with precisely \textbf{\emph{two}} hidden layer neurons! Moreover, the large $d$ converging values are observed to be in excellent agrement with the statistical physics replica theory based predictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05696 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed width treelike neural networks capacity analysis -- generic activations Stojnic, Mihailo Machine Learning Disordered Systems and Neural Networks Information Theory Probability We consider the capacity of \emph{treelike committee machines} (TCM) neural networks. Relying on Random Duality Theory (RDT), \cite{Stojnictcmspnncaprdt23} recently introduced a generic framework for their capacity analysis. An upgrade based on the so-called \emph{partially lifted} RDT (pl RDT) was then presented in \cite{Stojnictcmspnncapliftedrdt23}. Both lines of work focused on the networks with the most typical, \emph{sign}, activations. Here, on the other hand, we focus on networks with other, more general, types of activations and show that the frameworks of \cite{Stojnictcmspnncaprdt23,Stojnictcmspnncapliftedrdt23} are sufficiently powerful to enable handling of such scenarios as well. In addition to the standard \emph{linear} activations, we uncover that particularly convenient results can be obtained for two very commonly used activations, namely, the \emph{quadratic} and \emph{rectified linear unit (ReLU)} ones. In more concrete terms, for each of these activations, we obtain both the RDT and pl RDT based memory capacities upper bound characterization for \emph{any} given (even) number of the hidden layer neurons, $d$. In the process, we also uncover the following two, rather remarkable, facts: 1) contrary to the common wisdom, both sets of results show that the bounding capacity decreases for large $d$ (the width of the hidden layer) while converging to a constant value; and 2) the maximum bounding capacity is achieved for the networks with precisely \textbf{\emph{two}} hidden layer neurons! Moreover, the large $d$ converging values are observed to be in excellent agrement with the statistical physics replica theory based predictions. |
| title | Fixed width treelike neural networks capacity analysis -- generic activations |
| topic | Machine Learning Disordered Systems and Neural Networks Information Theory Probability |
| url | https://arxiv.org/abs/2402.05696 |