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arXiv · 1906.07709

Approximation power of random neural networks

Bolton Bailey·Ziwei Ji·Matus Telgarsky·Ruicheng Xian
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Abstract

This paper investigates the approximation power of three types of random neural networks: (a) infinite width networks, with weights following an arbitrary distribution; (b) finite width networks obtained by subsampling the preceding infinite width networks; (c) finite width networks obtained by starting with standard Gaussian initialization, and then adding a vanishingly small correction to the weights. The primary result is a fully quantified bound on the rate of approximation of general general continuous functions: in all three cases, a function $f$ can be approximated with complexity $\|f\|_1 (d/\delta)^{\mathcal{O}(d)}$, where $\delta$ depends on continuity properties of $f$ and the complexity measure depends on the weight magnitudes and/or cardinalities. Along the way, a variety of ancillary results are developed: an exact construction of Gaussian densities with infinite width networks, an elementary stand-alone proof scheme for approximation via convolutions of radial basis functions, subsampling rates for infinite width networks, and depth separation for corrected networks.

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Bolton Bailey, Ziwei Ji, Matus Telgarsky, Ruicheng Xian. 2019-06-18. Approximation power of random neural networks. https://arxiv.org/abs/1906.07709

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