Depth Separation for Neural Networks

Abstract: Let $$f:\mathbb{S}^{d-1}\times \mathbb{S}<sup>{d-1}\to\mathbb{S}$$ be a function of the form $f(\mathbf{x},\mathbf{x}&#39;) = g(\langle\mathbf{x},\mathbf{x}&#39;\rangle)$ 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 $g(x)=\sin(\pi d<sup>3x)$, the number of neurons must be $2<sup>{\Omega\left(d\log(d)\right)}$. Furthermore, the result holds w.r.t.\ the uniform distribution on $$\mathbb{S}^{d-1}\times \mathbb{S}<sup>{d-1}$$. 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 $$\mathbb{S}^{d-1}\times \mathbb{S}<sup>{d-1}$$.