# Learning from Synthetic Data: Limitations of ERM

arXiv: https://arxiv.org/abs/2601.15468 · HTML: https://mlsanity.com/paper/2601.15468/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in LLM synthetic data generation, one of 13 directions in Synthetic data and self-improvement for LLMs.
- Builds on 14 of its 20 references that are in the map.
- First posted on arXiv 21 Jan 2026.

## Placed in

- Synthetic data and self-improvement for LLMs / [LLM synthetic data generation](https://mlsanity.com/synth/llm-synthetic-data-generation/)

## Builds on (most cited first, 14 of 14)

- [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)
- [On the Stability of Iterative Retraining of Generative Models on their own Data](https://mlsanity.com/paper/2310.00429/) (2023)
- [How Bad is Training on Synthetic Data? A Statistical Analysis of Language Model Collapse](https://mlsanity.com/paper/2404.05090/) (2024)
- [Will Large-scale Generative Models Corrupt Future Datasets?](https://mlsanity.com/paper/2211.08095/) (2022)
- [Collapse or Thrive? Perils and Promises of Synthetic Data in a Self-Generating World](https://mlsanity.com/paper/2410.16713/) (2024)
- [Beyond Model Collapse: Scaling Up with Synthesized Data Requires Verification](https://mlsanity.com/paper/2406.07515/) (2024)
- [Self-Consuming Generative Models with Curated Data Provably Optimize Human Preferences](https://mlsanity.com/paper/2407.09499/) (2024)
- [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)
- [When Models Don't Collapse: On the Consistency of Iterative MLE](https://mlsanity.com/paper/2505.19046/) (2025)
- [Escaping Collapse: The Strength of Weak Data for Large Language Model Training](https://mlsanity.com/paper/2502.08924/) (2025)
- [Maximizing the Potential of Synthetic Data: Insights from Random Matrix Theory](https://mlsanity.com/paper/2410.08942/) (2024)
- [Learning from positive and unlabeled examples -Finite size sample bounds](https://mlsanity.com/paper/2507.07354/) (2025)

## Built on by (most cited first, 3 of 3)

- [Learning from Synthetic Data without Model Collapse in Iterative Instruction Tuning](https://mlsanity.com/paper/2607.17043/) (2026)
- [Pushing the Limits of High-Resolution Weather Forecasting through Data Scaling](https://mlsanity.com/paper/2608.14652/) (2026)
- [Minimax bounds for watermarked and masked recursive discrete distribution estimation](https://mlsanity.com/paper/2608.31091/) (2026)

## Abstract (arXiv)

The prevalence and low cost of LLMs have led to a rise of synthetic content. From review sites to court documents, "natural" content has been contaminated by data points that appear similar to natural data, but are in fact LLM-generated. In this work we revisit fundamental learning theory questions in this, now ubiquitous, setting. We model this scenario as a sequence of learning tasks where the input is a mix of natural and synthetic data, and the learning algorithms are oblivious to the origin of any individual example. We study the possibilities and limitations of ERM in this setting. For the problem of estimating the mean of an arbitrary d-dimensional distribution, we find that while ERM converges to the true mean, it is outperformed by an algorithm that assigns non-uniform weights to examples from different generations of data. For the PAC learning setting, the disparity is even more stark. We find that ERM does not always converge to the true concept, echoing the model collapse literature. However, we show there are algorithms capable of learning the correct hypothesis for arbitrary VC classes and arbitrary amounts of contamination.