Successfully and Efficiently Training Deep Multi-layer Perceptrons with Logistic Activation Function Simply Requires Initializing the Weights with an Appropriate Negative Mean

The vanishing gradient problem (i.e., gradients prematurely becoming extremely small during training, thereby effectively preventing a network from learning) is a long-standing obstacle to the training of deep neural networks using sigmoid activation functions when using the standard back-propagation algorithm. In this paper, we found that an important contributor to the problem is weight initialization. We started by developing a simple theoretical model showing how the expected value of gradients is affected by the mean of the initial weights. We then developed a second theoretical model that allowed us to identify a sufficient condition for the vanishing gradient problem to occur. Using these theories we found that initial back-propagation gradients do not vanish if the mean of the initial weights is negative and inversely proportional to the number of neurons in a layer. Numerous experiments with networks with 10 and 15 hidden layers corroborated the theoretical predictions: if we initialized weights as indicated by the theory, the standard back-propagation algorithm was both highly successful and efficient at training deep neural networks using sigmoid activation functions