Methods, systems, and apparatus, including computer programs encoded on a computer storage medium, for training a neural network that has one or more batch normalized neural network layers for use by a quantized inference system. One of the methods includes receiving a first batch of training data; determining batch normalization statistics for the first batch of training data; determining a correction factor from the batch normalization statistics for the first batch of training data and the long-term moving averages of the batch normalization statistics; generating batch normalized weights from the floating point weights for the batch normalized first neural network layer, comprising applying the correction factor to the floating point weights of the batch normalized first neural network layer; quantizing the batch normalized weights; determining a gradient of an objective function; and updating the floating point weights using the gradient.