文章基本信息

标题：Neural Networks with Small Weights and Depth-Separation Barriers
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作者：Gal Vardi ; Ohad Shamir
期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2020
卷号：2020
页码：1-39
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing results are limited to depths 2 and 3, and achieving results for higher depths has been an important open question. In this paper, we focus on feedforward ReLU networks, and prove fundamental barriers to proving such results beyond depth 4, by reduction to open problems and natural-proof barriers in circuit complexity. To show this, we study a seemingly unrelated problem of independent interest: Namely, whether there are polynomially-bounded functions which require super-polynomial weights in order to approximate with constant-depth neural networks. We provide a negative and constructive answer to that question, by showing that if a function can be approximated by a polynomially-sized, constant depth k network with arbitrarily large weights, it can also be approximated by a polynomially-sized, depth 3k 3 network, whose weights are polynomially bounded.