Near-optimal estimates for the $\ell^p$-Lipschitz constants of deep random ReLU neural networks

This paper studies the $\ell^p$-Lipschitz constants of ReLU neural networks $\Phi: \mathbb{R}^d \to \mathbb{R}$ with random parameters for $p \in [1,\infty]$. The distribution of the weights follows a variant of the He initialization and the biases are drawn from symmetric distributions. We derive high probability upper and lower bounds for wide networks that differ at most by a factor that is logarithmic in the network's width and linear in its depth. In the special case of shallow networks, we obtain matching bounds. Remarkably, the behavior of the $\ell^p$-Lipschitz constant varies significantly between the regimes $ p \in [1,2) $ and $ p \in [2,\infty] $. For $p \in [2,\infty]$, the $\ell^p$-Lipschitz constant behaves similarly to $\Vert g\Vert_{p'}$, where $g \in \mathbb{R}^d$ is a $d$-dimensional standard Gaussian vector and $1/p + 1/p' = 1$. In contrast, for $p \in [1,2)$, the $\ell^p$-Lipschitz constant aligns more closely to $\Vert g \Vert_{2}$.

On best approximation by multivariate ridge functions with applications to generalized translation networks

We prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n h_k(A_k x) \in \mathbb{R}$ with $h_k : \mathbb{R}^\ell \to \mathbb{R}$ and $A_k \in \mathbb{R}^{\ell \times d}$. We show that the order of approximation asymptotically behaves as $n^{-r/(d-\ell)}$, where $r$ is the regularity of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\infty$-Sobolev functions of regularity $r$ with error measured in $L^1$, while our upper bound applies to the approximation of $L^p$-Sobolev functions in $L^p$ for any $1 \leq p \leq \infty$. These bounds generalize well-known results about the approximation properties of univariate ridge functions to the multivariate case. Moreover, we use these bounds to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks.

On the Lipschitz constant of random neural networks

Empirical studies have widely demonstrated that neural networks are highly sensitive to small, adversarial perturbations of the input. The worst-case robustness against these so-called adversarial examples can be quantified by the Lipschitz constant of the neural network. However, only few theoretical results regarding this quantity exist in the literature. In this paper, we initiate the study of the Lipschitz constant of random ReLU neural networks, i.e., neural networks whose weights are chosen at random and which employ the ReLU activation function. For shallow neural networks, we characterize the Lipschitz constant up to an absolute numerical constant. Moreover, we extend our analysis to deep neural networks of sufficiently large width where we prove upper and lower bounds for the Lipschitz constant. These bounds match up to a logarithmic factor that depends on the depth.

Universal approximation with complex-valued deep narrow neural networks

We study the universality of complex-valued neural networks with bounded widths and arbitrary depths. Under mild assumptions, we give a full description of those activation functions $\varrho:\mathbb{C}\to \mathbb{C}$ that have the property that their associated networks are universal, i.e., are capable of approximating continuous functions to arbitrary accuracy on compact domains. Precisely, we show that deep narrow complex-valued networks are universal if and only if their activation function is neither holomorphic, nor antiholomorphic, nor $\mathbb{R}$-affine. This is a much larger class of functions than in the dual setting of arbitrary width and fixed depth. Unlike in the real case, the sufficient width differs significantly depending on the considered activation function. We show that a width of $2n+2m+5$ is always sufficient and that in general a width of $\max\{2n,2m\}$ is necessary. We prove, however, that a width of $n+m+4$ suffices for a rich subclass of the admissible activation functions. Here, $n$ and $m$ denote the input and output dimensions of the considered networks.

Optimal approximation of $C^k$-functions using shallow complex-valued neural networks

We prove a quantitative result for the approximation of functions of regularity $C^k$ (in the sense of real variables) defined on the complex cube $\Omega_n := [-1,1]^n +i[-1,1]^n\subseteq \mathbb{C}^n$ using shallow complex-valued neural networks. Precisely, we consider neural networks with a single hidden layer and $m$ neurons, i.e., networks of the form $z \mapsto \sum_{j=1}^m \sigma_j \cdot \phi\big(\rho_j^T z + b_j\big)$ and show that one can approximate every function in $C^k \left( \Omega_n; \mathbb{C}\right)$ using a function of that form with error of the order $m^{-k/(2n)}$ as $m \to \infty$, provided that the activation function $\phi: \mathbb{C} \to \mathbb{C}$ is smooth but not polyharmonic on some non-empty open set. Furthermore, we show that the selection of the weights $\sigma_j, b_j \in \mathbb{C}$ and $\rho_j \in \mathbb{C}^n$ is continuous with respect to $f$ and prove that the derived rate of approximation is optimal under this continuity assumption. We also discuss the optimality of the result for a possibly discontinuous choice of the weights.