High Probability Derivative Bounds for Random tanh Neural Networks on a Hypercube
We establish high-probability bounds for mixed input derivatives of wide random neural networks whose activation derivatives satisfy a factorial growth bound. Our main result specializes these estimates to $\tanh$ networks with Xavier initialization. A direct deterministic analysis based on Euclidean operator norms of the weight matrices yields derivative bounds that generally grow exponentially with the depth. We show that this growth can be substantially improved for sufficiently wide Gaussian networks by isolating the term that is linear in the highest-order derivative and controlling the corresponding tangent directions by measurable finite nets. For scalar-output $\tanh$ networks with Gaussian weights and Xavier initialization, we prove that there exist constants $C,C_0,C_1>0$ such that, whenever the common hidden width satisfies $n \geq C\left(L^3n_0^2(1+\log n_0)+L^2\left(1+\log(L/\eta)\right)\right)$, then, with probability at least $1-\eta$, the estimate $\left|D^u\mathcal{R}_{\Phi^{(L)}}(x)\right| \leq C_0 |u|! (C_1L)^{|u|-1}\prod_{j\in u}\beta_j(\eta,n_0)$ holds simultaneously for every non-empty $u\subseteq[n_0]$ and every $x\in[0,1]^{n_0}$. Thus, the first-order derivative bound is independent of the depth, while a square-free mixed derivative of order $|u|$ grows at most polynomially as $L^{|u|-1}$, apart from the coordinate factors. As consequences, we obtain high-probability bounds for the Euclidean Lipschitz constant and for weighted Sobolev norms of the network realization. The latter connect the derivative estimates to quasi-Monte Carlo integration and indicate how such regularity can enter the analysis of QMC-based training.