标题： 用神经网络逼近自然函数时的深度-宽度权衡 显示英文标题

标题： Depth-Width Tradeoffs in Approximating Natural Functions with Neural Networks

摘要： 我们为前馈神经网络提供了几个基于深度的分离结果，证明了使用更深的网络而非更浅的网络可以更好地近似各种类型的简单且自然的函数，即使较浅的网络规模大得多。 这包括球体和椭圆体的指示函数；相对于$L_1$范数具有径向对称性的非线性函数；以及平滑的非线性函数。 我们还表明，这些差距可以在实验中观察到：当训练神经网络学习单位球体的指示函数时，增加深度确实比增加宽度能实现更好的学习效果。 显示英文摘要

摘要： We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-linear functions which are radial with respect to the $L_1$ norm; and smooth non-linear functions. We also show that these gaps can be observed experimentally: Increasing the depth indeed allows better learning than increasing width, when training neural networks to learn an indicator of a unit ball.