Abstract: A computer-implemented method of training a deep neural network, comprising, for each of one or more batches of training examples: processing the data in a forward pass through the layers of the network, by: applying a set of network weights to the input data to obtain a set of weighted inputs, normalising the weighted inputs based on statistics computed for each training example, transforming the normalised inputs by affine transformation parameters, applying an activation function to the transformed normalised inputs to obtain post-activation values, and normalizing the post-activation values based on one or more proxy variables sampled from a distribution defined by proxy distribution parameters, the normalization applied independently of training example; processing the data in a backward pass through the network to determine updates to learnable parameters comprising network weights, affine transformation parameters, and proxy distribution parameters, and updating the learnable parameters to optimise a p