Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons

• URL: http://arxiv.org/abs/2107.02397v2
• Date: Wed, 7 Jul 2021 18:21:55 GMT
• Title: Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons
• Authors: Zuowei Shen and Haizhao Yang and Shijun Zhang
• Abstract summary: We develop feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. We prove that $sigma$-activated networks with width $36d(2d+1)$ and depth $11$ can approximate any continuous function on a $d$-dimensioanl hypercube within an arbitrarily small error.
• Score: 5.37133760455631
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a simple and computable continuous activation function $\sigma$ leveraging a triangular-wave function and a softsign function. We prove that $\sigma$-activated networks with width $36d(2d+1)$ and depth $11$ can approximate any continuous function on a $d$-dimensioanl hypercube within an arbitrarily small error. Hence, for supervised learning and its related regression problems, the hypothesis space generated by these networks with a size not smaller than $36d(2d+1)\times 11$ is dense in the space of continuous functions. Furthermore, classification functions arising from image and signal classification are in the hypothesis space generated by $\sigma$-activated networks with width $36d(2d+1)$ and depth $12$, when there exist pairwise disjoint closed bounded subsets of $\mathbb{R}^d$ such that the samples of the same class are located in the same subset.

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