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# Depth Separation for Neural Networks

*Proceedings of the 2017 Conference on Learning Theory*, PMLR 65:690-696, 2017.

#### Abstract

Let $f:\mathbb{S}^d-1\times \mathbb{S}^d-1\to\mathbb{S}$ be a function of the form $f(x,x’) = g(⟨x,x’⟩)$ for $g:[-1,1]\to \mathbb{R}$. We give a simple proof that shows that poly-size depth two neural networks with (exponentially) bounded weights cannot approximate $f$ whenever $g$ cannot be approximated by a low degree polynomial. Moreover, for many $g$’s, such as $g(x)=\sin(\pi d^3x)$, the number of neurons must be $2^Ω\left(d\log(d)\right)$. Furthermore, the result holds w.r.t. the uniform distribution on $\mathbb{S}^d-1\times \mathbb{S}^d-1$. As many functions of the above form can be well approximated by poly-size depth three networks with poly-bounded weights, this establishes a separation between depth two and depth three networks w.r.t. the uniform distribution on $\mathbb{S}^d-1\times \mathbb{S}^d-1$.