Constructive Deep ReLU Neural Network Approximation

We propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f : [ - 1 , 1 ] K → R . We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and...

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Published in:	Journal of scientific computing Vol. 90; no. 2; p. 75
Main Authors:	Herrmann, Lukas, Opschoor, Joost A. A., Schwab, Christoph
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.02.2022 Springer Nature B.V
Subjects:	Accuracy Algorithms Approximation Artificial neural networks Computational Mathematics and Numerical Analysis Convergence Fourier transforms Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Neural networks Optimization Partial differential equations Polynomials Theoretical Exponential convergence Deep ReLU neural networks 41A10 65D05 Neural network construction Generalization error 41A50 65D15
ISSN:	0885-7474, 1573-7691
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Abstract	We propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f : [ - 1 , 1 ] K → R . We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small Lebesgue constants, and by a-priori (i.e., “offline”) emulation of a spectral basis with DNNs to prescribed fidelity. Assuming availability of N function values of a possibly corrupted, numerical approximation f ˘ of f in [ - 1 , 1 ] K and a bound on ‖ f - f ˘ ‖ L ∞ ( [ - 1 , 1 ] K ) , we provide an explicit, computational construction of a ReLU DNN which attains accuracy ε (depending on N and ‖ f - f ˘ ‖ L ∞ ( [ - 1 , 1 ] K ) ) uniformly, with respect to the inputs. For analytic maps f : [ - 1 , 1 ] K → R , we prove exponential convergence of expression and generalization errors of the constructed ReLU DNNs. Specifically, for every target accuracy ε ∈ ( 0 , 1 ) , there exists N depending also on f such that the error of the construction algorithm with N evaluations of f ˘ as input in the norm L ∞ ( [ - 1 , 1 ] K ; R ) is smaller than ε up to an additive data-corruption bound ‖ f - f ˘ ‖ L ∞ ( [ - 1 , 1 ] K ) multiplied with a factor growing slowly with 1 / ε and the number of non-zero DNN weights grows polylogarithmically with respect to 1 / ε . The algorithmic construction of the ReLU DNNs which will realize the approximations, is explicit and deterministic in terms of the function values of f ˘ in tensorized Clenshaw–Curtis grids in [ - 1 , 1 ] K . We illustrate the proposed methodology by a constructive algorithm for (offline) computations of posterior expectations in Bayesian PDE inversion.
AbstractList	We propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f:[-1,1]K→R. We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small Lebesgue constants, and by a-priori (i.e., “offline”) emulation of a spectral basis with DNNs to prescribed fidelity. Assuming availability of N function values of a possibly corrupted, numerical approximation f˘ of f in [-1,1]K and a bound on ‖f-f˘‖L∞([-1,1]K), we provide an explicit, computational construction of a ReLU DNN which attains accuracy ε (depending on N and ‖f-f˘‖L∞([-1,1]K)) uniformly, with respect to the inputs. For analytic maps f:[-1,1]K→R, we prove exponential convergence of expression and generalization errors of the constructed ReLU DNNs. Specifically, for every target accuracy ε∈(0,1), there exists N depending also on f such that the error of the construction algorithm with N evaluations of f˘ as input in the norm L∞([-1,1]K;R) is smaller than ε up to an additive data-corruption bound ‖f-f˘‖L∞([-1,1]K) multiplied with a factor growing slowly with 1/ε and the number of non-zero DNN weights grows polylogarithmically with respect to 1/ε. The algorithmic construction of the ReLU DNNs which will realize the approximations, is explicit and deterministic in terms of the function values of f˘ in tensorized Clenshaw–Curtis grids in [-1,1]K. We illustrate the proposed methodology by a constructive algorithm for (offline) computations of posterior expectations in Bayesian PDE inversion. We propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f : [ - 1 , 1 ] K → R . We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small Lebesgue constants, and by a-priori (i.e., “offline”) emulation of a spectral basis with DNNs to prescribed fidelity. Assuming availability of N function values of a possibly corrupted, numerical approximation f ˘ of f in [ - 1 , 1 ] K and a bound on ‖ f - f ˘ ‖ L ∞ ( [ - 1 , 1 ] K ) , we provide an explicit, computational construction of a ReLU DNN which attains accuracy ε (depending on N and ‖ f - f ˘ ‖ L ∞ ( [ - 1 , 1 ] K ) ) uniformly, with respect to the inputs. For analytic maps f : [ - 1 , 1 ] K → R , we prove exponential convergence of expression and generalization errors of the constructed ReLU DNNs. Specifically, for every target accuracy ε ∈ ( 0 , 1 ) , there exists N depending also on f such that the error of the construction algorithm with N evaluations of f ˘ as input in the norm L ∞ ( [ - 1 , 1 ] K ; R ) is smaller than ε up to an additive data-corruption bound ‖ f - f ˘ ‖ L ∞ ( [ - 1 , 1 ] K ) multiplied with a factor growing slowly with 1 / ε and the number of non-zero DNN weights grows polylogarithmically with respect to 1 / ε . The algorithmic construction of the ReLU DNNs which will realize the approximations, is explicit and deterministic in terms of the function values of f ˘ in tensorized Clenshaw–Curtis grids in [ - 1 , 1 ] K . We illustrate the proposed methodology by a constructive algorithm for (offline) computations of posterior expectations in Bayesian PDE inversion.
ArticleNumber	75
Author	Herrmann, Lukas Opschoor, Joost A. A. Schwab, Christoph
Author_xml	– sequence: 1 givenname: Lukas orcidid: 0000-0003-3402-6420 surname: Herrmann fullname: Herrmann, Lukas email: lukas.herrmann@ricam.oeaw.ac.at organization: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences – sequence: 2 givenname: Joost A. A. surname: Opschoor fullname: Opschoor, Joost A. A. organization: Seminar for Applied Mathematics, ETH Zürich – sequence: 3 givenname: Christoph surname: Schwab fullname: Schwab, Christoph organization: Seminar for Applied Mathematics, ETH Zürich
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SubjectTerms	Accuracy Algorithms Approximation Artificial neural networks Computational Mathematics and Numerical Analysis Convergence Fourier transforms Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Neural networks Optimization Partial differential equations Polynomials Theoretical
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Title	Constructive Deep ReLU Neural Network Approximation
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