# Heat Death of Generative Models in Closed-Loop Learning

arXiv: https://arxiv.org/abs/2404.02325 · HTML: https://mlsanity.com/paper/2404.02325/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 11 of its 15 references that are in the map.
- First posted on arXiv 2 Apr 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 11 of 11)

- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [A Survey of Large Language Models](https://mlsanity.com/paper/2303.18223/) (2023)
- [WILDS: A Benchmark of in-the-Wild Distribution Shifts](https://mlsanity.com/paper/2012.07421/) (2020)
- [The Curse of Recursion: Training on Generated Data Makes Models Forget](https://mlsanity.com/paper/2305.17493/) (2023)
- [Poisoning Web-Scale Training Datasets is Practical](https://mlsanity.com/paper/2302.10149/) (2023)
- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [A Survey on Generative Diffusion Model](https://mlsanity.com/paper/2209.02646/) (2022)
- [On the Stability of Iterative Retraining of Generative Models on their own Data](https://mlsanity.com/paper/2310.00429/) (2023)
- [Towards Understanding the Interplay of Generative Artificial Intelligence and the Internet](https://mlsanity.com/paper/2306.06130/) (2023)
- [Combining Generative Artificial Intelligence (AI) and the Internet: Heading towards Evolution or Degradation?](https://mlsanity.com/paper/2303.01255/) (2023)
- [Analysis of stochastic approximation schemes with set-valued maps in the absence of a stability guarantee and their stabilization](https://mlsanity.com/paper/1701.07590/) (2017)

## Built on by (most cited first, 22 of 22)

- [Collapse or Thrive? Perils and Promises of Synthetic Data in a Self-Generating World](https://mlsanity.com/paper/2410.16713/) (2024)
- [Position: Model Collapse Does Not Mean What You Think](https://mlsanity.com/paper/2503.03150/) (2025)
- [Universality of the π2/6 Pathway in Avoiding Model Collapse](https://mlsanity.com/paper/2410.22812/) (2024)
- [Rate of Model Collapse in Recursive Training](https://mlsanity.com/paper/2412.17646/) (2024)
- [A Theoretical Perspective: How to Prevent Model Collapse in Self-consuming Training Loops](https://mlsanity.com/paper/2502.18865/) (2025)
- [When Models Don't Collapse: On the Consistency of Iterative MLE](https://mlsanity.com/paper/2505.19046/) (2025)
- [Model Collapse in the Self-Consuming Chain of Diffusion Finetuning: A Novel Perspective from Quantitative Trait Modeling](https://mlsanity.com/paper/2407.17493/) (2024)
- [What happens when generative AI models train recursively on each others' outputs?](https://mlsanity.com/paper/2505.21677/) (2025)
- [Meanings and Feelings of Large Language Models: Observability of Latent States in Generative AI](https://mlsanity.com/paper/2405.14061/) (2024)
- [Language Generation with Replay: A Learning-Theoretic View of Model Collapse](https://mlsanity.com/paper/2603.11784/) (2026)
- [Recursive InPainting (RIP): how much information is lost under recursive inferences?](https://mlsanity.com/paper/2407.09549/) (2024)
- [Interacting Large Language Model Agents. Interpretable Models and Social Learning](https://mlsanity.com/paper/2411.01271/) (2024)
- [Convergence Dynamics and Stabilization Strategies of Co-Evolving Generative Models](https://mlsanity.com/paper/2503.08117/) (2025)
- [Context-Selective State Space Models: Feedback is All You Need](https://mlsanity.com/paper/2510.14027/) (2025)
- [GenCtrl -- A Formal Controllability Toolkit for Generative Models](https://mlsanity.com/paper/2601.05637/) (2026)
- [A theoretical basis for model collapse in recursive training](https://mlsanity.com/paper/2506.09401/) (2025)
- [Why Self-Rewarding Works: Theoretical Guarantees for Iterative Alignment of Language Models](https://mlsanity.com/paper/2601.22513/) (2026)
- [Lost in Retraining: Roaming the Parameter Space of Exponential Families Under Closed-Loop Learning](https://mlsanity.com/paper/2506.20623/) (2025)
- [Adynamical systems view of training generativemodels and the memorization phenomenon](https://mlsanity.com/paper/2605.19483/) (2026)
- [Minimax bounds for watermarked and masked recursive discrete distribution estimation](https://mlsanity.com/paper/2608.31091/) (2026)
- [Preventing Model Collapse: A Fisher-Rao Perspective on the Dynamics of Training with Synthetic Data](https://mlsanity.com/paper/2609.18878/) (2026)
- [Self-Generated Feedback Destabilizes Test-Time Training: A Causal Decomposition of Long-Horizon Adaptation](https://mlsanity.com/paper/2610.05076/) (2026)

## Abstract (arXiv)

Improvement and adoption of generative machine learning models is rapidly accelerating, as exemplified by the popularity of LLMs (Large Language Models) for text, and diffusion models for image generation. As generative models become widespread, data they generate is incorporated into shared content through the public web. This opens the question of what happens when data generated by a model is fed back to the model in subsequent training campaigns. This is a question about the stability of the training process, whether the distribution of publicly accessible content, which we refer to as "knowledge", remains stable or collapses. Small scale empirical experiments reported in the literature show that this closed-loop training process is prone to degenerating. Models may start producing gibberish data, or sample from only a small subset of the desired data distribution (a phenomenon referred to as mode collapse). So far there has been only limited theoretical understanding of this process, in part due to the complexity of the deep networks underlying these generative models. The aim of this paper is to provide insights into this process (that we refer to as "generative closed-loop learning") by studying the learning dynamics of generative models that are fed back their own produced content in addition to their original training dataset. The sampling of many of these models can be controlled via a "temperature" parameter. Using dynamical systems tools, we show that, unless a sufficient amount of external data is introduced at each iteration, any non-trivial temperature leads the model to asymptotically degenerate. In fact, either the generative distribution collapses to a small set of outputs or becomes uniform over a large set of outputs.