标题： 通过动力系统视角的神经网络深度-宽度权衡优化 显示英文标题

标题： Better Depth-Width Trade-offs for Neural Networks through the lens of Dynamical Systems

摘要： 神经网络的表达能力作为其深度、宽度和激活单元类型的函数，一直是深度学习理论中的一个重要问题。 最近，通过与动力系统的新联系，利用连续映射 $f$的广义固定点概念，称为周期点，获得了 ReLU 网络的深度分离结果。 在本工作中，我们加强了与动力系统的联系，并在多个方面改进了现有的宽度下界。 我们的第一个主要结果是在更强的 $L^1$-近似误差概念下成立的特定周期宽度下界，而不是较弱的分类误差。 我们的第二个贡献是，在先前结果不适用的区域中，我们提供了更紧的宽度下界，仍然产生有意义的指数深度-宽度分离。 我们结果的一个副产品是，只要 $f$具有奇数周期，就存在一个表征深度-宽度权衡的通用常数。 技术上，我们的结果是通过揭示给定函数的以下三个量之间的更紧密联系得出的：其周期、其利普希茨常数以及该函数 $f$与自身复合时产生的振荡数量的增长率。 显示英文摘要

摘要： The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a continuous map $f$, called periodic points. In this work, we strengthen the connection with dynamical systems and we improve the existing width lower bounds along several aspects. Our first main result is period-specific width lower bounds that hold under the stronger notion of $L^1$-approximation error, instead of the weaker classification error. Our second contribution is that we provide sharper width lower bounds, still yielding meaningful exponential depth-width separations, in regimes where previous results wouldn't apply. A byproduct of our results is that there exists a universal constant characterizing the depth-width trade-offs, as long as $f$ has odd periods. Technically, our results follow by unveiling a tighter connection between the following three quantities of a given function: its period, its Lipschitz constant and the growth rate of the number of oscillations arising under compositions of the function $f$ with itself.