# From Collapse to Improvement: Statistical Perspectives on the Evolutionary Dynamics of Iterative Training on Contaminated Sources

arXiv: https://arxiv.org/abs/2602.10531 · HTML: https://mlsanity.com/paper/2602.10531/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Model collapse from synthetic data, one of 13 directions in Synthetic data and self-improvement for LLMs.
- Builds on 26 of its 42 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Placed in

- Synthetic data and self-improvement for LLMs / [Model collapse from synthetic data](https://mlsanity.com/synth/model-collapse-from-synthetic-data/)

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [LAION-5B: An open large-scale dataset for training next generation image-text models](https://mlsanity.com/paper/2210.08402/) (2022)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [A Survey on Evaluation of Large Language Models](https://mlsanity.com/paper/2307.03109/) (2023)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [The Curse of Recursion: Training on Generated Data Makes Models Forget](https://mlsanity.com/paper/2305.17493/) (2023)
- [Self-Consuming Generative Models Go MAD](https://mlsanity.com/paper/2307.01850/) (2023)
- [Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data](https://mlsanity.com/paper/2404.01413/) (2024)
- [A Tale of Tails: Model Collapse as a Change of Scaling Laws](https://mlsanity.com/paper/2402.07043/) (2024)
- [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)
- [How Bad is Training on Synthetic Data? A Statistical Analysis of Language Model Collapse](https://mlsanity.com/paper/2404.05090/) (2024)
- [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)
- [Self-Consuming Generative Models with Curated Data Provably Optimize Human Preferences](https://mlsanity.com/paper/2407.09499/) (2024)
- [Towards Theoretical Understandings of Self-Consuming Generative Models](https://mlsanity.com/paper/2402.11778/) (2024)
- [Nepotistically Trained Generative-AI Models Collapse](https://mlsanity.com/paper/2311.12202/) (2023)
- [The Rise of AI-Generated Content in Wikipedia](https://mlsanity.com/paper/2410.08044/) (2024)
- [Regurgitative Training: The Value of Real Data in Training Large Language Models](https://mlsanity.com/paper/2407.12835/) (2024)
- [A Training-free Method for LLM Text Attribution](https://mlsanity.com/paper/2501.02406/) (2025)

## Built on by (most cited first, 1 of 1)

- [Prediction-Only Distillation in Linear and Logistic Regression](https://mlsanity.com/paper/2607.15450/) (2026)

## Abstract (arXiv)

The problem of model collapse has presented new challenges in iterative training of generative models, where such training with synthetic data leads to an overall degradation of performance. This paper looks at the problem from a statistical viewpoint, illustrating that one can actually hope for improvement when models are trained on data contaminated with synthetic samples, as long as there is some amount of fresh information from the true target distribution. In particular, we consider iterative training on samples sourced from a mixture of the true target and synthetic distributions. We analyze the entire iterative evolution in a next-token prediction language model, capturing how the interplay between the mixture weights and the sample size controls the overall long-term performance. With non-trivial mixture weight of the true distribution, even if it decays over time, simply training the model in a contamination-agnostic manner with appropriate sample sizes can avoid collapse and even recover the true target distribution under certain conditions. Simulation studies support our findings and also show that such behavior is more general for other classes of models.