MLPs at the EOC: Concentration of the NTK

Abstract

We study the concentration of the Neural Tangent Kernel (NTK) $K_{\theta }:{\mathbb{R}}^{m_0}\times {\mathbb{R}}^{m_0}\to {\mathbb{R}}^{m_l\times m_l}$ of $l$-layer Multilayer Perceptrons (MLPs) $N:{\mathbb{R}}^{m_0}\times \Theta \to {\mathbb{R}}^{m_l}$ equipped with activation functions $\varphi (s)=as+b|s|$ for some $a,b\in \mathbb{R}$ with the parameter $\theta \in \Theta$ being initialized at the Edge Of Chaos (EOC). Without relying on the gradient independence assumption that has only been shown to hold asymptotically in the infinitely wide limit, we prove that an approximate version of gradient independence holds at finite width. Showing that the NTK entries $K_{\theta }(x_{i_1},x_{i_2})$ for $i_1,i_2\in [1:n]$ over a dataset $\{x_1,\cdots ,x_n\}\subset {\mathbb{R}}^{m_0}$ concentrate simultaneously via maximal inequalities, we prove that the NTK matrix $K(\theta )=[\frac{1}{n}{K}_{\theta }(x_{i_1},x_{i_2}):i_1,i_2\in [1:n]]\in {\mathbb{R}}^{n{m}_l\times n{m}_l}$ concentrates around its infinitely wide limit $\stackrel{\infty }{K}\in {\mathbb{R}}^{n{m}_l\times n{m}_l}$ without the need for linear overparameterization. Our results imply that in order to accurately approximate the limit, hidden layer widths have to grow quadratically as $m_k=k^{2}m$ for some $m\in \mathbb{N}+1$ for sufficient concentration. For such MLPs, we obtain the concentration bound $\mathbb{P}(\Vert K(\theta )-\stackrel{\infty }{K}\Vert \le O(({\Delta }_{\varphi }^{-2}+m_l^{\frac{1}{2}}l){\kappa }_{\varphi }^2{m}^{-\frac{1}{2}}))\ge 1-O(m^{-1})$ modulo logarithmic terms, where we denoted ${\Delta }_{\varphi }=\frac{b^2}{a^2+b^2}$ and ${\kappa }_{\varphi }=\frac{|a|+|b|}{\sqrt{a^2+b^2}}$. This reveals in particular that the absolute value (${\Delta }_{\varphi }=1$, ${\kappa }_{\varphi }=1$) beats the ReLU (${\Delta }_{\varphi }=\frac{1}{2}$, ${\kappa }_{\varphi }=\sqrt{2}$) in terms of the concentration of the NTK.

arXiv comment36 pages, 1 figure