PredProp: Bidirectional Stochastic Optimization with Precision Weighted Predictive Coding

Abstract

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 Pred-Prop 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.

Backpropagation at the Infinitesimal Inference Limit of Energy-Based Models: Unifying Predictive Coding, Equilibrium Propagation, and Contrastive Hebbian Learning

“…The learning rules used by PC networks (PCNs) require only local and Hebbian updates (Millidge, Tschantz, Seth & Buckley, 2020b) and a variety of neural microcircuits have been proposed that can implement the computations required by PC (Bastos et al, 2012;Keller & Mrsic-Flogel, 2018). Moreover, recent works have begun exploring the use of large-scale PCNs in machine learning tasks, to some success (Kinghorn, Millidge & Buckley, 2021;Lotter, Kreiman & Cox, 2016;Millidge, 2019;Ofner & Stober, 2021;Salvatori, Pinchetti et al, 2022;Salvatori et al, 2021). Unlike the other algorithms presented here, PC has a mathematical interpretation as in terms of variational Bayesian inference (Bogacz, 2017;Buckley, Kim, McGregor & Seth, 2017;Friston, 2003Friston, , 2005Millidge, Seth & Buckley, 2021), and the variables in the model can be mapped to explicit probabilistic elements of a generative model.…”