  There is an increasing convergence between biologically plausible
computational models of inference and learning with local update rules and the
global gradient-based optimization of neural network models employed in machine
learning. One particularly exciting connection is the correspondence between
the locally informed optimization in predictive coding networks and the error
backpropagation algorithm that is used to train state-of-the-art deep
artificial neural networks. Here we focus on the related, but still largely
under-explored connection between precision weighting in predictive coding
networks and the Natural Gradient Descent algorithm for deep neural networks.
Precision-weighted predictive coding is an interesting candidate for scaling up
uncertainty-aware optimization -- particularly for models with large parameter
spaces -- due to its distributed nature of the optimization process and the
underlying local approximation of the Fisher information metric, the adaptive
learning rate that is central to Natural Gradient Descent. Here, we show that
hierarchical predictive coding networks with learnable precision indeed are
able to solve various supervised and unsupervised learning tasks with
performance comparable to global backpropagation with natural gradients and
outperform their classical gradient descent counterpart on tasks where high
amounts of noise are embedded in data or label inputs. When applied to
unsupervised auto-encoding of image inputs, the deterministic network produces
hierarchically organized and disentangled embeddings, hinting at the close
connections between predictive coding and hierarchical variational inference.
