On the Dynamics of Inference and Learning

TL;DR

This paper models Bayesian inference as a continuous dynamical system, analyzing its behavior and learning rates, and compares it to neural network training dynamics on benchmark datasets.

Contribution

It introduces a differential equation framework for Bayesian updating, linking inference dynamics to information geometry and neural network training.

Findings

Learning rate follows a 1/T power-law when the Cramér-Rao bound is saturated.

Hidden variables add a driving term to the inference flow equation.

Neural network training exhibits similar power-law behavior in final loss regimes.

Abstract

Statistical Inference is the process of determining a probability distribution over the space of parameters of a model given a data set. As more data becomes available this probability distribution becomes updated via the application of Bayes' theorem. We present a treatment of this Bayesian updating process as a continuous dynamical system. Statistical inference is then governed by a first order differential equation describing a trajectory or flow in the information geometry determined by a parametric family of models. We solve this equation for some simple models and show that when the Cram\'{e}r-Rao bound is saturated the learning rate is governed by a simple $1/T$ power-law, with $T$ a time-like variable denoting the quantity of data. The presence of hidden variables can be incorporated in this setting, leading to an additional driving term in the resulting flow equation. We illustrate this with both analytic and numerical examples based on Gaussians and Gaussian Random Processes and inference of the coupling constant in the 1D Ising model. Finally we compare the qualitative behaviour exhibited by Bayesian flows to the training of various neural networks on benchmarked data sets such as MNIST and CIFAR10 and show how that for networks exhibiting small final losses the simple power-law is also satisfied.