Differentiable Generalised Predictive Coding

This paper deals with differentiable dynamical models congruent with neural process theories that cast brain function as the hierarchical refinement of an internal generative model explaining observations. Our work extends existing implementations of gradient-based predictive coding with automatic differentiation and allows to integrate deep neural networks for non-linear state parameterization. Gradient-based predictive coding optimises inferred states and weights locally in for each layer by optimising precision-weighted prediction errors that propagate from stimuli towards latent states. Predictions flow backwards, from latent states towards lower layers. The model suggested here optimises hierarchical and dynamical predictions of latent states. Hierarchical predictions encode expected content and hierarchical structure. Dynamical predictions capture changes in the encoded content along with higher order derivatives. Hierarchical and dynamical predictions interact and address different aspects of the same latent states. We apply the model to various perception and planning tasks on sequential data and show their mutual dependence. In particular, we demonstrate how learning sampling distances in parallel address meaningful locations data sampled at discrete time steps. We discuss possibilities to relax the assumption of linear hierarchies in favor of more flexible graph structure with emergent properties. We compare the granular structure of the model with canonical microcircuits describing predictive coding in biological networks and review the connection to Markov Blankets as a tool to characterize modularity. A final section sketches out ideas for efficient perception and planning in nested spatio-temporal hierarchies.

PredProp: Bidirectional Stochastic Optimization with Precision Weighted Predictive Coding

We present PredProp, a method for bidirectional, parallel and local optimisation of weights, activities and precision in neural networks. PredProp jointly addresses inference and learning, scales learning rates dynamically and weights gradients by the curvature of the loss function by optimizing prediction error precision. PredProp optimizes network parameters with Stochastic Gradient Descent and error forward propagation based strictly on prediction errors and variables locally available to each layer. Neighboring layers optimise shared activity variables so that prediction errors can propagate forward in the network, while predictions propagate backwards. This process minimises the negative Free Energy, or evidence lower bound of the entire network. We show that networks trained with PredProp resemble gradient based predictive coding when the number of weights between neighboring activity variables is one. In contrast to related work, PredProp generalizes towards backward connections of arbitrary depth and optimizes precision for any deep network architecture. Due to the analogy between prediction error precision and the Fisher information for each layer, PredProp implements a form of Natural Gradient Descent. When optimizing DNN models, layer-wise PredProp renders the model a bidirectional predictive coding network. Alternatively DNNs can parameterize the weights between two activity variables. We evaluate PredProp for dense DNNs on simple inference, learning and combined tasks. We show that, without an explicit sampling step in the network, PredProp implements a form of variational inference that allows to learn disentangled embeddings from low amounts of data and leave evaluation on more complex tasks and datasets to future work.