  Compositionality is a basic structural feature of both biological and
artificial neural networks. Learning compositional functions via gradient
descent incurs well known problems like vanishing and exploding gradients,
making careful learning rate tuning essential for real-world applications. This
paper proves that multiplicative weight updates satisfy a descent lemma
tailored to compositional functions. Based on this lemma, we derive Madam -- a
multiplicative version of the Adam optimiser -- and show that it can train
state of the art neural network architectures without learning rate tuning. We
further show that Madam is easily adapted to train natively compressed neural
networks by representing their weights in a logarithmic number system. We
conclude by drawing connections between multiplicative weight updates and
recent findings about synapses in biology.
