arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 2003.00777cs.LGmath.DSstat.ML
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通过动力系统的透镜更好地权衡神经网络的深度-宽度

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

Stanford University(斯坦福大学);Singapore University of Technology and Design(新加坡科技设计大学)

Vaggos Chatziafratis, Sai Ganesh Nagarajan, Ioannis Panageas

摘要:

神经网络的表达能力是其深度、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,ReLU网络的深度分离结果是通过与动力系统的新连接获得的,使用了连续映射$f$的不动点的广义概念,称为周期点。在这项工作中,我们加强了与动力系统的联系,我们从几个方面改进了现有的宽度下界。我们的第一个主要结果是特定于周期的宽度下限,持有较强的概念下的$L^1$近似误差,而不是较弱的分类错误。我们的第二个贡献是,我们提供了更清晰的宽度下限,仍然产生有意义的指数深度-宽度分离,在以前的结果不适用的制度。我们的结果的一个副产品是,存在一个普适常数表征的深度-宽度的权衡,只要$f$有奇数周期。从技术上讲,我们的结果揭示了以下三个量之间的一个给定的函数:它的周期,它的Lipschitz常数和增长率的振荡下产生的组合物的功能$f$与本身的数量更紧密的连接。

英文摘要:

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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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