Implicit regularization of dropout

It is important to understand how the popular regularization method dropout helps the neural network training find a good generalization solution. In this work, we theoretically derive the implicit regularization of dropout and study the relation between the Hessian matrix of the loss function and the covariance matrix of the dropout noise, supported by a series of experiments. We then numerically study two implications of the implicit regularization of dropout, which intuitively rationalize why dropout helps generalization. First, we find that the training with dropout finds the neural network with a flatter minimum compared with standard gradient descent training in experiments, and the implicit regularization is the key for finding flat solutions. Second, trained with dropout, input weights of hidden neurons (the input weight of a hidden neuron consists of the weight from its input layer to the hidden neuron and its bias term) would tend to condense on isolated orientations. Condensation is a feature in non-linear learning process, which makes the neural network low complexity. Although our theory mainly focuses on the dropout used in the last hidden layer, our experiments apply for general dropout in training neural networks. This work points out the distinct characteristics of dropout compared with stochastic gradient descent and serves as an important basis for fully understanding dropout.