  Direct Feedback Alignment (DFA) is emerging as an efficient and biologically
plausible alternative to the ubiquitous backpropagation algorithm for training
deep neural networks. Despite relying on random feedback weights for the
backward pass, DFA successfully trains state-of-the-art models such as
Transformers. On the other hand, it notoriously fails to train convolutional
networks. An understanding of the inner workings of DFA to explain these
diverging results remains elusive. Here, we propose a theory for the success of
DFA. We first show that learning in shallow networks proceeds in two steps: an
alignment phase, where the model adapts its weights to align the approximate
gradient with the true gradient of the loss function, is followed by a
memorisation phase, where the model focuses on fitting the data. This two-step
process has a degeneracy breaking effect: out of all the low-loss solutions in
the landscape, a network trained with DFA naturally converges to the solution
which maximises gradient alignment. We also identify a key quantity underlying
alignment in deep linear networks: the conditioning of the alignment matrices.
The latter enables a detailed understanding of the impact of data structure on
alignment, and suggests a simple explanation for the well-known failure of DFA
to train convolutional neural networks. Numerical experiments on MNIST and
CIFAR10 clearly demonstrate degeneracy breaking in deep non-linear networks and
show that the align-then-memorise process occurs sequentially from the bottom
layers of the network to the top.
