Figure 2.
Equilibrium Propagation. The framework considers networks with dynamics described by the minimisation of an energy function. As the activity of these networks converges to an equilibrium, the energy simultaneously decays (blue arrows) to a minimum given the current weights. Once in equilibrium, the weighs are modified (green arrows). It has been shown that network error can be minimised if the synaptic weights are modified in two steps (schematically illustrated by the two displays in the top box; [22]). First, with only the input pattern provided, once the network converges, weights are modified in the direction in which the energy increases. Second, the output layer is additionally constrained to values closer to the target pattern (particular details described in [22]). Constraining the output nodes changes the energy landscape for the units in the middle layers. Once these units converge to a new equilibrium, weights are modified in the direction in which the energy decreases. Scellier and Bengio [22] noted that for temporal-error networks, this procedure gives the contrastive learning rule (Equation 2.2). The predictive coding networks, however, converge to an equilibrium in the first step where the free-energy function reaches its global minimum [14]; thus, there is no weight modification required by the equilibrium propagation framework. Therefore, only a single phase (i.e., the second phase) and a single weight update are required in the explicit-error models, and it only involves Hebbian plasticity.