We present a theoretical and empirical study of the gradient dynamics of
overparameterized shallow ReLU networks with one-dimensional input, solving
least-squares interpolation. We show that the gradient dynamics of such
networks are determined by the gradient flow in a non-redundant
parameterization of the network function. We examine the principal qualitative
features of this gradient flow. In particular, we determine conditions for two
learning regimes:kernel and adaptive, which depend both on the relative
magnitude of initialization of weights in different layers and the asymptotic
behavior of initialization coefficients in the limit of large network widths.
We show that learning in the kernel regime yields smooth interpolants,
minimizing curvature, and reduces to cubic splines for uniform initializations.
Learning in the adaptive regime favors instead linear splines, where knots
cluster adaptively at the sample points.

本文介绍了在一个维度输入下，解决最小二乘插值的超参数化浅层 ReLU 网络的梯度动态的理论和实证研究，阐述了这种网络的梯度动态通过网络函数的非冗余参数化来确定，同时还研究了两种学习模式：核函数和自适应函数，以及它们所对应的性质。

浅层一元 ReLU 网络的梯度动力学

Gradient Dynamics of Shallow Univariate ReLU Networks

A recent line of work studies overparametrized neural networks in the "kernel
regime," i.e. when the network behaves during training as a kernelized linear
predictor, and thus training with gradient descent has the effect of finding
the minimum RKHS norm solution. This stands in contrast to other studies which
demonstrate how gradient descent on overparametrized multilayer networks can
induce rich implicit biases that are not RKHS norms. Building on an observation
by Chizat and Bach, we show how the scale of the initialization controls the
transition between the "kernel" (aka lazy) and "rich" (aka active) regimes and
affects generalization properties in multilayer homogeneous models. We provide
a complete and detailed analysis for a simple two-layer model that already
exhibits an interesting and meaningful transition between the kernel and rich
regimes, and we demonstrate the transition for more complex matrix
factorization models and multilayer non-linear networks.

研究发现神经网络在过参量化后，当行为符合核化线性预测器时，在梯度下降的训练下能找到最小 RKHS 范数解；与此不同的是，梯度下降在过参量化的多层网络上可能引入不是 RKHS 范数的丰富隐性偏差。本文基于 Chizat 和 Bach 的观察，展示了初始化规模是如何控制多层同质模型中的 “核”（也称惰性）和 “丰富”（也称活动）区域之间的转换并影响泛化属性的。我们对一个简单的两层模型进行了全面而详细的分析，已经展示了核与丰富区域之间有一个有趣和有意义的转换，并展示了矩阵分解模型和多层非线性网络的转换。