Optimization Variance:  Exploring Generalization Properties of DNNs
Authors: Xiao Zhang, Dongrui Wu, Haoyi Xiong, Bo Dai
Keywords: Double Descent, Neural Networks, Generalization
Abstract: Unlike the conventional wisdom in statistical learning theory, the test error of a deep neural network (DNN) often demonstrates double descent: as the model complexity increases, it first follows a classical U-shaped curve and then shows a second descent. Through bias-variance decomposition, recent studies revealed that the bell-shaped variance is the major cause of model-wise double descent (when the DNN is widened gradually). This paper investigates epoch-wise double descent, i.e., the test error of a DNN also shows double descent as the number of training epochs increases. Specifically, we extend the bias-variance analysis to epoch-wise double descent, and reveal that the variance also contributes the most to the zero-one loss, as in model-wise double descent. Inspired by this result, we propose a novel metric called optimization variance to measure the diversity of model updates caused by the stochastic gradients of random training batches drawn in the same iteration. This metric can be estimated using samples from the training set only but correlates well with the test error. The proposed optimization variance can be used to predict the generalization ability of a DNN, and hence early stopping can be achieved without using any validation set.