  Deep learning, a multi-layered neural network approach inspired by the brain,
has revolutionized machine learning. One of its key enablers has been
backpropagation, an algorithm that computes the gradient of a loss function
with respect to the weights in the neural network model, in combination with
its use in gradient descent. However, the implementation of deep learning in
digital computers is intrinsically wasteful, with energy consumption becoming
prohibitively high for many applications. This has stimulated the development
of specialized hardware, ranging from neuromorphic CMOS integrated circuits and
integrated photonic tensor cores to unconventional, material-based computing
systems. The learning process in these material systems, taking place, e.g., by
artificial evolution or surrogate neural network modelling, is still a
complicated and time-consuming process. Here, we demonstrate an efficient and
accurate homodyne gradient extraction method for performing gradient descent on
the loss function directly in the material system. We demonstrate the method in
our recently developed dopant network processing units, where we readily
realize all Boolean gates. This shows that gradient descent can in principle be
fully implemented in materio using simple electronics, opening up the way to
autonomously learning material systems.
