[Paper Review] Predictive Coding Approximates Backprop along Arbitrary Computation Graphs

Beren Millidge, Alexander Tschantz+1 more|arXiv (Cornell University)|Jun 7, 2020

Advanced Memory and Neural Computing17 citations

TL;DR

This paper demonstrates that predictive coding, a biologically plausible learning framework relying on local and Hebbian plasticity, asymptotically approximates backpropagation through arbitrary computation graphs. It shows that predictive coding networks can train complex models like CNNs, RNNs, and LSTMs with performance equivalent to backprop, enabling biologically realistic implementations of standard deep learning architectures.

ABSTRACT

Backpropagation of error (backprop) is a powerful algorithm for training machine learning architectures through end-to-end differentiation. However, backprop is often criticised for lacking biological plausibility. Recently, it has been shown that backprop in multilayer-perceptrons (MLPs) can be approximated using predictive coding, a biologically-plausible process theory of cortical computation which relies only on local and Hebbian updates. The power of backprop, however, lies not in its instantiation in MLPs, but rather in the concept of automatic differentiation which allows for the optimisation of any differentiable program expressed as a computation graph. Here, we demonstrate that predictive coding converges asymptotically (and in practice rapidly) to exact backprop gradients on arbitrary computation graphs using only local learning rules. We apply this result to develop a straightforward strategy to translate core machine learning architectures into their predictive coding equivalents. We construct predictive coding CNNs, RNNs, and the more complex LSTMs, which include a non-layer-like branching internal graph structure and multiplicative interactions. Our models perform equivalently to backprop on challenging machine learning benchmarks, while utilising only local and (mostly) Hebbian plasticity. Our method raises the potential that standard machine learning algorithms could in principle be directly implemented in neural circuitry, and may also contribute to the development of completely distributed neuromorphic architectures.

Motivation & Objective

• To bridge the gap between biologically plausible learning and the powerful optimization capabilities of backpropagation in deep learning.
• To demonstrate that predictive coding can approximate automatic differentiation across arbitrary computation graphs, not just multilayer perceptrons.
• To develop a systematic method for translating standard machine learning architectures into predictive coding equivalents using only local connectivity and Hebbian plasticity.
• To validate that predictive coding networks achieve performance comparable to backprop in real-world benchmarks while maintaining biological plausibility.

Proposed method

• The authors use variational inference and free-energy minimization to frame predictive coding as an optimization of a generative model over a computation graph.
• They derive variational update rules for the mean and variance of latent variables, with the variance having a closed-form solution that simplifies optimization.
• The dynamics of the system are modeled as gradient descent on the free-energy, with the update rule for each node depending only on local prediction errors and local derivatives.
• Fixed-point equations are derived to show that the system converges to a solution equivalent to backpropagation gradients in the limit.
• The method is applied to construct predictive coding equivalents of CNNs, RNNs, and LSTMs, including models with multiplicative interactions and non-layer-like structures.
• The approach uses only local connectivity and Hebbian-like learning rules, with no need for weight sharing or non-local error transmission.

Experimental results

Research questions

• RQ1 Can predictive coding approximate backpropagation not only in multilayer perceptrons but also in arbitrary computation graphs?
• RQ2 Can the learning rules in predictive coding networks be strictly local and Hebbian while still achieving gradients equivalent to backprop?
• RQ3 Can predictive coding be used to train complex architectures like LSTMs and CNNs with performance matching standard backprop training?
• RQ4 Is there a formal mathematical equivalence between predictive coding and automatic differentiation in general computation graphs?
• RQ5 Can this framework support uncertainty-aware learning through learnable precisions in the generative model?

Key findings

• Predictive coding converges asymptotically to exact backpropagation gradients on arbitrary computation graphs using only local and Hebbian learning rules.
• The method successfully constructs predictive coding equivalents of CNNs, RNNs, and LSTMs, including models with complex internal branching and multiplicative interactions.
• The predictive coding models achieve performance on par with standard backprop-trained models on challenging machine learning benchmarks.
• The derivation shows that the optimal variational variance has a closed-form solution, simplifying the optimization process and allowing focus on mean updates.
• The fixed-point equations derived from the free-energy minimization process exactly recover the backpropagation update rules in the limit, confirming mathematical equivalence.
• The framework supports uncertainty-aware learning through learnable precisions, enabling a form of attention to prediction uncertainty during training.