Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing Its Gradient Estimator Bias

Abstract

Equilibrium Propagation is a biologically-inspired algorithm that trains convergent recurrent neural networks with a local learning rule. This approach constitutes a major lead to allow learning-capable neuromophic systems and comes with strong theoretical guarantees. Equilibrium propagation operates in two phases, during which the network is let to evolve freely and then “nudged” toward a target; the weights of the network are then updated based solely on the states of the neurons that they connect. The weight updates of Equilibrium Propagation have been shown mathematically to approach those provided by Backpropagation Through Time (BPTT), the mainstream approach to train recurrent neural networks, when nudging is performed with infinitely small strength. In practice, however, the standard implementation of Equilibrium Propagation does not scale to visual tasks harder than MNIST. In this work, we show that a bias in the gradient estimate of equilibrium propagation, inherent in the use of finite nudging, is responsible for this phenomenon and that canceling it allows training deep convolutional neural networks. We show that this bias can be greatly reduced by using symmetric nudging (a positive nudging and a negative one). We also generalize Equilibrium Propagation to the case of cross-entropy loss (by opposition to squared error). As a result of these advances, we are able to achieve a test error of 11.7% on CIFAR-10, which approaches the one achieved by BPTT and provides a major improvement with respect to the standard Equilibrium Propagation that gives 86% test error. We also apply these techniques to train an architecture with unidirectional forward and backward connections, yielding a 13.2% test error. These results highlight equilibrium propagation as a compelling biologically-plausible approach to compute error gradients in deep neuromorphic systems.

Neurons learn by predicting future activity

“…Temporal difference learning is one of the most promising ideas about how backpropagation-like algorithms could be implemented in the brain. It is based on using differences in neuronal activity to approximate top-down error signals 4,[18][19][20][21][22][23][24] . A typical example of such algorithms is contrastive Hebbian learning [25][26][27] , which was proven to be equivalent to backpropagation under certain assumptions 28 .…”

Neurons learn by predicting future activity

“…For pooling, we used the max pooling with 2x2 filters and stride 2. The activation function for the convolutional and the fully connected layers was the hard-sigmoid activation function, S(x) = (1+hardtanh(x-1))*0.5, as implemented in (26). The learning rates were: 0.4, 0.028, and 0.025 for the first, second convolutional layer, and for the fully connected output layer, respectively.…”

“…Temporal difference learning is one of the most promising ideas of how backpropagation-like algorithms could be implemented in the brain. It is based on using differences in neuronal activity to approximate top-down error signals (4,(20)(21)(22)(23)(24)(25)(26). A typical example of such algorithms is Contrastive Hebbian Learning (27)(28)(29), which was proven to be equivalent to backpropagation under certain assumptions (30).…”

Photonic kernel machine learning for ultrafast spectral analysis

“…Such architectures come with their own technical difficulties. Because they rely on co-located memory and processing resources, standard optimization algorithms can prove impractical, although novel optimization procedures have recently been put forward to overcome this obstacle by meeting their peculiar hardware design [77][78][79][80][81].…”