Predictive Coding with Approximate Laplace Monte Carlo

Umais Zahid, Qinghai Guo, Karl Friston, Zafeirios Fountas

Keywords: predictive coding, variational Bayes, Laplace approximation, generative modelling, free energy

TL;DR: A novel method that improves the performance of predictive coding by incorporating information about the curvature of the energy landscape.

Abstract: Predictive coding (PC) accounts of perception now form one of the dominant computational theories of the brain. Despite this, they have enjoyed little export to the broader field of machine learning, where comparative generative models have flourished. In part, this has been due to the poor performance of models trained with standard implementations of PC when evaluated by both sample quality and marginal likelihood. By adopting the perspective of PC as a variational Bayes algorithm under the Laplace approximation, we identify the source of these deficits to lie in the exclusion of an associated Hessian term in the standard PC objective function. To remedy this, we make three primary contributions: we begin by suggesting a simple Monte Carlo estimated evidence lower bound which relies on sampling from the Hessian-parameterised variational posterior. We then derive a novel block diagonal approximation to the full Hessian matrix that has lower memory requirements and favourable mathematical properties. Lastly, we present an algorithm that combines our method with standard PC to reduce memory complexity further. We evaluate models trained with our approach against the standard PC framework on image benchmark datasets. Our approach produces higher log-likelihoods and qualitatively better samples that more closely capture the diversity of the data-generating distribution.

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