On the Global Convergence of Gradient Descent for multi-layer ResNets in the mean-field regime
This addresses the theoretical understanding of why deep ResNets converge to global optima despite nonconvexity, which is incremental as it builds on prior mean-field analyses.
The paper tackles the nonconvex optimization problem of training multi-layer ResNets by analyzing gradient descent convergence in the mean-field regime, showing that with sufficient network size, first-order methods can find global minimizers that fit training data.
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.