  Equilibrium Propagation (EP) is a biologically inspired learning algorithm
for convergent recurrent neural networks, i.e. RNNs that are fed by a static
input x and settle to a steady state. Training convergent RNNs consists in
adjusting the weights until the steady state of output neurons coincides with a
target y. Convergent RNNs can also be trained with the more conventional
Backpropagation Through Time (BPTT) algorithm. In its original formulation EP
was described in the case of real-time neuronal dynamics, which is
computationally costly. In this work, we introduce a discrete-time version of
EP with simplified equations and with reduced simulation time, bringing EP
closer to practical machine learning tasks. We first prove theoretically, as
well as numerically that the neural and weight updates of EP, computed by
forward-time dynamics, are step-by-step equal to the ones obtained by BPTT,
with gradients computed backward in time. The equality is strict when the
transition function of the dynamics derives from a primitive function and the
steady state is maintained long enough. We then show for more standard
discrete-time neural network dynamics that the same property is approximately
respected and we subsequently demonstrate training with EP with equivalent
performance to BPTT. In particular, we define the first convolutional
architecture trained with EP achieving ~ 1% test error on MNIST, which is the
lowest error reported with EP. These results can guide the development of deep
neural networks trained with EP.
