Lost in Retraining: Roaming the Parameter Space of Exponential Families Under Closed-Loop Learning

TL;DR

This paper investigates the dynamics of parameter estimation in exponential family models under closed-loop learning, revealing how initial biases can be amplified or mitigated through different estimation strategies.

Contribution

It derives equations of motion for model parameters in closed-loop learning for exponential families and analyzes conditions affecting convergence and bias amplification.

Findings

Maximum likelihood estimation leads to convergence to bias-amplifying states.

Presence of ground truth data prevents bias amplification with MAP or regularisation.

The process's dynamics are governed by equations of motion derived for exponential families.

Abstract

Closed-loop learning is the process of repeatedly estimating a model from data generated from the model itself. It is receiving great attention due to the possibility that large neural network models may, in the future, be primarily trained with data generated by artificial neural networks themselves. We study this process for models that belong to exponential families, deriving equations of motions that govern the dynamics of the parameters. We show that maximum likelihood estimation of the parameters endows sufficient statistics with the martingale property and that as a result the process converges to absorbing states that amplify initial biases present in the data. However, we show that this outcome may be prevented if the data contains at least one data point generated from a ground truth model, by relying on maximum a posteriori estimation or by introducing regularisation.