Abstract
We introduce Equilibrium Propagation, a learning framework for energy-based models. It involves only one kind of neural computation, performed in both the first phase (when the prediction is made) and the second phase of training (after the target or prediction error is revealed). Although this algorithm computes the gradient of an objective function just like Backpropagation, it does not need a special computation or circuit for the second phase, where errors are implicitly propagated. Equilibrium Propagation shares similarities with Contrastive Hebbian Learning and Contrastive Divergence while solving the theoretical issues of both algorithms: our algorithm computes the gradient of a well-defined objective function. Because the objective function is defined in terms of local perturbations, the second phase of Equilibrium Propagation corresponds to only nudging the prediction (fixed point or stationary distribution) toward a configuration that reduces prediction error. In the case of a recurrent multi-layer supervised network, the output units are slightly nudged toward their target in the second phase, and the perturbation introduced at the output layer propagates backward in the hidden layers. We show that the signal “back-propagated” during this second phase corresponds to the propagation of error derivatives and encodes the gradient of the objective function, when the synaptic update corresponds to a standard form of spike-timing dependent plasticity. This work makes it more plausible that a mechanism similar to Backpropagation could be implemented by brains, since leaky integrator neural computation performs both inference and error back-propagation in our model. The only local difference between the two phases is whether synaptic changes are allowed or not. We also show experimentally that multi-layer recurrently connected networks with 1, 2, and 3 hidden layers can be trained by Equilibrium Propagation on the permutation-invariant MNIST task.
Highlights
The Backpropagation algorithm to train neural networks is considered to be biologically implausible
Backpropagation applies to any differentiable computational graph, Equilibrium Propagation applies to a whole class of energy based models
In the second phase of the training framework, when the target values for output units are observed, the outputs are nudged toward their targets and the network relaxes to a new but nearby fixed point which corresponds to slightly smaller prediction error
Summary
The Backpropagation algorithm to train neural networks is considered to be biologically implausible. This framework applies to a whole class of energy based models, which is not limited to the continuous Hopfield model but encompasses arbitrary dynamics whose fixed points (or stationary distributions) correspond to minima of an energy function. Our algorithm computes the gradient of a sound objective function that corresponds to local perturbations It can be realized with leaky integrator neural computation which performs both inference (in the first phase) and back-propagation of error derivatives (in the second phase). The code for the model is available for replicating and extending the experiments
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