On the Global Convergence of Gradient Descent for multi-layer ResNets in the mean-field regime

• Main Authors: Ding, Zhiyan, Chen, Shi, Li, Qin, Wright, Stephen
• Format: Journal Article
• Language: English
• Published: 06.10.2021
• Subjects: Computer Science - Learning; Computer Science - Numerical Analysis; Mathematics - Numerical Analysis; Statistics - Machine Learning
• DOI: 10.48550/arxiv.2110.02926

Finding the optimal configuration of parameters in ResNet is a nonconvex minimization problem, but first-order methods nevertheless find the global optimum in the overparameterized regime. We study this phenomenon with mean-field analysis, by translating the training process of ResNet to a gradient-flow partial differential equation (PDE) and examining the convergence properties of this limiting process. The activation function is assumed to be $2$-homogeneous or partially $1$-homogeneous; the regularized ReLU satisfies the latter condition. We show that if the ResNet is sufficiently large, with depth and width depending algebraically on the accuracy and confidence levels, first-order optimization methods can find global minimizers that fit the training data.