Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation

We introduce Equilibrium Propagation (e-prop), a learning algorithm for energy-based models. This algorithm involves only one kind of neural computation both for the first phase (when the prediction is made) and the second phase (after the target is revealed) of training. Contrary to backpropagation in feedforward networks, there is no need for special computation in the second phase of our learning algorithm. Equilibrium Propagation combines features of Contrastive Hebbian Learning and Contrastive Divergence while solving the theoretical issues of both algorithms: the algorithm computes the exact gradient of a well defined objective function. Because the objective function is defined in terms of local perturbations, the second phase of e-prop corresponds to only nudging the first-phase fixed point towards a configuration that has lower cost value. In the case of a multi-layer supervised neural network, the output units are slightly nudged towards their target, and the perturbation introduced at the output layer propagates backward in the network. The theory developed in this paper shows that the signal 'back-propagated' during this second phase actually contains information about the error derivatives, which we use to implement a learning rule proved to perform gradient descent with respect to the objective function. Thus, this work makes it more plausible that a mechanism similar to backpropagation could be implemented by brains.