In deep learning (DL) the instability phenomenon is widespread and well documented, most commonly using the classical measure of stability, the Lipschitz constant. While a small Lipchitz constant is traditionally viewed as guarantying stability, it does not capture the instability phenomenon in DL for classification well. The reason is that a classification function -- which is the target function to be approximated -- is necessarily discontinuous, thus having an 'infinite' Lipchitz constant. As a result, the classical approach will deem every classification function unstable, yet basic classification functions a la 'is there a cat in the image?' will typically be locally very 'flat' -- and thus locally stable -- except at the decision boundary. The lack of an appropriate measure of stability hinders a rigorous theory for stability in DL, and consequently, there are no proper approximation theoretic results that can guarantee the existence of stable networks for classification functions. In this paper we introduce a novel stability measure $\mathscr{S}(f)$, for any classification function $f$, appropriate to study the stability of discontinuous functions and their approximations. We further prove two approximation theorems: First, for any $\epsilon > 0$ and any classification function $f$ on a \emph{compact set}, there is a neural network (NN) $\psi$, such that $\psi - f \neq 0$ only on a set of measure $< \epsilon$, moreover, $\mathscr{S}(\psi) \geq \mathscr{S}(f) - \epsilon$ (as accurate and stable as $f$ up to $\epsilon$). Second, for any classification function $f$ and $\epsilon > 0$, there exists a NN $\psi$ such that $\psi = f$ on the set of points that are at least $\epsilon$ away from the decision boundary.

在深度学习中，我们引入了一种新的稳定性测量方法，适用于研究不连续函数及其逼近的稳定性。通过证明两个逼近定理，我们展示了对于任何分类函数，都可以构建一个神经网络使其在小于一定误差的测度内准确且稳定，同时也可以构建一个神经网络使其与分类函数在离决策边界至少一定距离的点上相等。

分类问题中是否存在稳定的神经网络？-- 人工智能稳定性的新视角