Online Training of Hopfield Networks using Predictive Coding

TL;DR

This work investigates applying Predictive Coding (PC) learning to recurrent Hopfield networks (HNs) to enable biologically plausible, local credit assignment without time unrolling. By constructing PC-HN architectures (a single 100-unit loop and a 50-30-20 three-layer loop) and training with PC update rules, the authors demonstrate memory recall behavior analogous to classical HNs, including recall from corrupted inputs and convergence to trained targets. Linear stability analyses confirm equilibria are typically stable, while perturbation experiments show robust recall, though some spurious equilibria and near-zero eigenvalues suggest opportunities for further refinement. The study highlights PC as a viable framework for training RNNs in a physically plausible, equilibrium-driven manner, with potential extensions to more complex recurrent architectures.

Abstract

Neuroscience and Artificial Intelligence (AI) have progressed in tandem, each contributing to our understanding of the brain, and inspiring recent developments in biologically-plausible neural networks (NNs) and learning rules. Predictive coding (PC), and its learning rule, have been shown to approximate error backpropagation in a biologically relevant manner, with local weight updates that depend only on the activity of the pre- and post-synaptic neurons. Unlike traditional feedforward NNs where the flow of information goes in one direction, PC models mimic the brain more accurately by passing information bidirectionally: prediction in one direction, and correction/error in the other. PC models learn by clamping some neurons to target values and running the network to equilibrium. At equilibrium, the network calculates its own error gradients right at the location where they are used for weight updates. Traditional backprop requires the computation graph to be feedforward. However, the PC version of backprop does not have this requirement. Amazingly, no one has demonstrated the application of PC learning directly to recurrent neural networks (RNNs). Hopfield networks (HNs) are RNNs that implement a content-addressable memory, learning patterns (or ``memories'') that can be retrieved from partial or corrupted patterns. In this paper, we show that a HN can be trained using the PC learning rules without modification. To our knowledge, this is the first time PC learning has been applied directly to train a RNN, without the need to unroll it in time. Our results indicate that the PC-trained HNs behave like classical HNs.

Figures (6)

• Figure 1: Part of a traditional PC network. Each circle represents a population of neurons.
• Figure 2: A single population of PC units, recurrently connected to itself. The circles represent populations of neurons. The arrows internal to the unit are one-to-one connections, while the connections that loop outside the unit are dense connections.
• Figure 3: A loop of three PC populations. Each PC population is not recurrently connected to itself, but the entire loop forms a recurrent network.
• Figure 4: Each plot shows the distance between the network state and the target patterns over time after being initialized with perturbed targets. This was done for each of the 10 targets the network was trained on (see legends) and in all cases, the the network converges to the target pattern. Results are presented from the 50-30-20 loop (top) and the single population of 100 units (bottom), and for real-valued targets (left) and binary targets (right).
• Figure 5: Ten runs showing convergence to the corresponding target, and not to any of the other targets. Each plot includes 10 lines of each colour, where the colour indicates the distance to one of the target patterns. Notice that the network converges to target 1 (green), and does not converge to any of the other targets. These plots show results for the 50-30-20 network, but results were similar for the single-population network, and for other targets.