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# (Non-)Convergence Results for Predictive Coding Networks

*Proceedings of the 39th International Conference on Machine Learning*, PMLR 162:6793-6810, 2022.

#### Abstract

Predictive coding networks (PCNs) are (un)supervised learning models, coming from neuroscience, that approximate how the brain works. One major open problem around PCNs is their convergence behavior. In this paper, we use dynamical systems theory to formally investigate the convergence of PCNs as they are used in machine learning. Doing so, we put their theory on a firm, rigorous basis, by developing a precise mathematical framework for PCN and show that for sufficiently small weights and initializations, PCNs converge for any input. Thereby, we provide the theoretical assurance that previous implementations, whose convergence was assessed solely by numerical experiments, can indeed capture the correct behavior of PCNs. Outside of the identified regime of small weights and small initializations, we show via a counterexample that PCNs can diverge, countering common beliefs held in the community. This is achieved by identifying a Neimark-Sacker bifurcation in a PCN of small size, which gives rise to an unstable fixed point and an invariant curve around it.