# Global convergence of ResNets: From finite to infinite width using linear parameterization

arXiv: https://arxiv.org/abs/2112.05531 · HTML: https://mlsanity.com/paper/2112.05531/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Dec 2021.

## Built on by (most cited first, 9 of 9)

- [Do Residual Neural Networks discretize Neural Ordinary Differential Equations?](https://mlsanity.com/paper/2205.14612/) (2022)
- [Implicit regularization of deep residual networks towards neural ODEs](https://mlsanity.com/paper/2309.01213/) (2023)
- [Understanding the training of infinitely deep and wide ResNets with Conditional Optimal Transport](https://mlsanity.com/paper/2403.12887/) (2024)
- [Field theory for optimal signal propagation in ResNets](https://mlsanity.com/paper/2305.07715/) (2023)
- [Global Convergence in Training Large-Scale Transformers](https://mlsanity.com/paper/2410.23610/) (2024)
- [Variational formulations of ODE-Net as a mean-field optimal control problem and existence results](https://mlsanity.com/paper/2303.05924/) (2023)
- [Generalization of Scaled Deep ResNets in the Mean-Field Regime](https://mlsanity.com/paper/2403.09889/) (2024)
- [Training-Induced Escape from Token Clustering in a Mean-Field Formulation of Transformers](https://mlsanity.com/paper/2605.07772/) (2026)
- [Dropout and Random Gradient Masking Are Asymptotically Equivalent in Large ResNets](https://mlsanity.com/paper/2607.16761/) (2026)

## Abstract (arXiv)

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.