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

arXiv: https://arxiv.org/abs/2506.20623 · HTML: https://mlsanity.com/paper/2506.20623/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 16 references that are in the map.
- First posted on arXiv 25 Jun 2025.

## Builds on (most cited first, 8 of 8)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [On the Stability of Iterative Retraining of Generative Models on their own Data](https://mlsanity.com/paper/2310.00429/) (2023)
- [The Curious Decline of Linguistic Diversity: Training Language Models on Synthetic Text](https://mlsanity.com/paper/2311.09807/) (2023)
- [Model Collapse Demystified: The Case of Regression](https://mlsanity.com/paper/2402.07712/) (2024)
- [Large Language Models Suffer From Their Own Output: An Analysis of the Self-Consuming Training Loop](https://mlsanity.com/paper/2311.16822/) (2023)
- [Heat Death of Generative Models in Closed-Loop Learning](https://mlsanity.com/paper/2404.02325/) (2024)
- [The Spectrum of Fisher Information of Deep Networks Achieving Dynamical Isometry](https://mlsanity.com/paper/2006.07814/) (2020)

## Abstract (arXiv)

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.