  Backpropagation through nonlinear neurons is an outstanding challenge to the
field of optical neural networks and the major conceptual barrier to
all-optical training schemes. Each neuron is required to exhibit a
directionally dependent response to propagating optical signals, with the
backwards response conditioned on the forward signal, which is highly
non-trivial to implement optically. We propose a practical and surprisingly
simple solution that uses saturable absorption to provide the network
nonlinearity. We find that the backward propagating gradients required to train
the network can be approximated in a pump-probe scheme that requires only
passive optical elements. Simulations show that, with readily obtainable
optical depths, our approach can achieve equivalent performance to
state-of-the-art computational networks on image classification benchmarks,
even in deep networks with multiple sequential gradient approximations. This
scheme is compatible with leading optical neural network proposals and
therefore provides a feasible path towards end-to-end optical training.
