A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths. In the finite-width setting, the network architectures we consider are fairly general: we require the existence of a wide layer with roughly order of $N$ neurons, $N$ being the number of data samples; and the scaling of the remaining layer widths is arbitrary (up to logarithmic factors). To obtain our results, we analyze various quantities of independent interest: we give lower bounds on the smallest singular value of hidden feature matrices, and upper bounds on the Lipschitz constant of input-output feature maps.
翻译:最近的一项工作通过Neural Tangent Kernel(NTK)分析了深神经网络的理论特性。 特别是,NTK最小的精度值与记忆能力、梯度下位算法的全球趋同和深网的普遍化有关,然而,现有的结果要么提供了两层设置的界限,要么假定NTK矩阵的频谱与多层网络的 0 相隔。在本文中,我们为深RELU网提供了NTK矩阵最小的精度值的严格界限,这在无限宽度和有限宽度的限制方面都是如此。 在有限的宽度设置中,我们认为网络结构相当笼统:我们需要一个大致为N$的宽层,$N$是数据样品的数量;以及剩余层宽度的扩大是任意的(加上对数因素)。 为了获得我们的结果,我们分析了各种独立兴趣的数量:我们对隐藏的地平面图最小的奇数值,我们给隐藏的地平面图的最小的奇数值,以及Lip的上框。