Overparameterization of deep ResNet: zero loss and mean-field analysis

• URL: http://arxiv.org/abs/2105.14417v1
• Date: Sun, 30 May 2021 02:46:09 GMT
• Title: Overparameterization of deep ResNet: zero loss and mean-field analysis
• Authors: Zhiyan Ding and Shi Chen and Qin Li and Stephen Wright
• Abstract summary: Finding parameters in a deep neural network (NN) that fit data is a non optimization problem. We show that a basic first-order optimization method (gradient descent) finds a global solution with perfect fit in many practical situations. We give estimates of the depth and width needed to reduce the loss below a given threshold, with high probability.
• Score: 19.45069138853531
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global solution with perfect fit in many practical situations. We examine this phenomenon for the case of Residual Neural Networks (ResNet) with smooth activation functions in a limiting regime in which both the number of layers (depth) and the number of neurons in each layer (width) go to infinity. First, we use a mean-field-limit argument to prove that the gradient descent for parameter training becomes a partial differential equation (PDE) that characterizes gradient flow for a probability distribution in the large-NN limit. Next, we show that the solution to the PDE converges in the training time to a zero-loss solution. Together, these results imply that training of the ResNet also gives a near-zero loss if the Resnet is large enough. We give estimates of the depth and width needed to reduce the loss below a given threshold, with high probability.

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