Overparametrization is a key factor in the absence of convexity to explain global convergence of gradient descent (GD) for neural networks. Beside the well studied lazy regime, infinite width (mean field) analysis has been developed for shallow networks, using on convex optimization technics. To bridge the gap between the lazy and mean field regimes, we study Residual Networks (ResNets) in which the residual block has linear parametrization while still being nonlinear. Such ResNets admit both infinite depth and width limits, encoding residual blocks in a Reproducing Kernel Hilbert Space (RKHS). In this limit, we prove a local Polyak-Lojasiewicz inequality. Thus, every critical point is a global minimizer and a local convergence result of GD holds, retrieving the lazy regime. In contrast with other mean-field studies, it applies to both parametric and non-parametric cases under an expressivity condition on the residuals. Our analysis leads to a practical and quantified recipe: starting from a universal RKHS, Random Fourier Features are applied to obtain a finite dimensional parameterization satisfying with high-probability our expressivity condition.

翻译：过参数化是解释神经网络梯度下降（GD）全局收敛性时，在缺乏凸性的情况下起关键作用的因素。除了被广泛研究的惰性机制外，基于凸优化技术，针对浅层网络已发展出无限宽度（平均场）分析方法。为弥合惰性机制与平均场机制之间的差距，我们研究了残差网络（ResNet），其中残差块采用线性参数化但仍保持非线性。此类ResNet同时支持无限深度与无限宽度极限，将残差块编码至再生核希尔伯特空间（RKHS）中。在此极限下，我们证明了局部Polyak-Lojasiewicz不等式成立。因此，每个临界点均为全局极小值点，且GD具有局部收敛性，从而复现了惰性机制。与其他平均场研究不同，该结论在残差满足表达性条件时，同时适用于参数化与非参数化情形。我们的分析得出了实用且量化的方案：从通用RKHS出发，应用随机傅里叶特征获得有限维参数化，该参数化以高概率满足我们的表达性条件。