  The implementation of artificial neural networks in hardware substrates is a
major interdisciplinary enterprise. Well suited candidates for physical
implementations must combine nonlinear neurons with dedicated and efficient
hardware solutions for both connectivity and training. Reservoir computing
addresses the problems related with the network connectivity and training in an
elegant and efficient way. However, important questions regarding impact of
reservoir size and learning routines on the convergence-speed during learning
remain unaddressed. Here, we study in detail the learning process of a recently
demonstrated photonic neural network based on a reservoir. We use a greedy
algorithm to train our neural network for the task of chaotic signals
prediction and analyze the learning-error landscape. Our results unveil
fundamental properties of the system's optimization hyperspace. Particularly,
we determine the convergence speed of learning as a function of reservoir size
and find exceptional, close to linear scaling. This linear dependence, together
with our parallel diffractive coupling, represent optimal scaling conditions
for our photonic neural network scheme.
