Global convergence of ResNets: From finite to infinite width using linear parameterization
This work addresses a foundational theoretical gap in deep learning convergence analysis, providing insights for researchers in optimization and neural network theory.
The paper tackles the problem of understanding global convergence of gradient descent for Residual Networks (ResNets) by bridging lazy and mean-field regimes, proving that in infinite width and depth limits with linear parameterization, every critical point is a global minimizer and local convergence holds, with a practical recipe using Random Fourier Features to achieve this with high probability.
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.