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    Smaller generalization error derived for deep compared to shallow residual neural networks

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    Type
    Preprint
    Authors
    Kammonen, Aku
    Kiessling, Jonas
    Plecháč, Petr
    Sandberg, Mattias
    Szepessy, Anders
    Tempone, Raul cc
    KAUST Department
    Applied Mathematics and Computational Science Program
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Stochastic Numerics Research Group
    Date
    2020-10-05
    Permanent link to this record
    http://hdl.handle.net/10754/665560
    
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    Abstract
    Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \text{Re}\sum_{k=1}^K\bar b_{\ell k}e^{{\rm i}\omega_{\ell k}\bar z_\ell}+ \text{Re}\sum_{k=1}^K\bar c_{\ell k}e^{{\rm i}\omega'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega_{\ell k},\omega'_{\ell k})$ of the random Fourier features $e^{{\rm i}\omega_{\ell k}\bar z_\ell}$ and $e^{{\rm i}\omega'_{\ell k}\cdot x}$ is derived. The derivation is based on the corresponding generalization error to approximate function values $f(x)$. The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(LK)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes $LK$, in the case the $L^\infty$-norm of $f$ is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network that shows promising results.
    Publisher
    arXiv
    arXiv
    2010.01887
    Additional Links
    https://arxiv.org/pdf/2010.01887
    Collections
    Preprints; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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