Learning from Synthetic Data: Limitations of ERM

TL;DR

This work studies learning from data where natural and AI-generated samples are mixed and the origin of each example is unknown, modeling contamination with a parameter $\alpha$. It shows exact variance for the empirical mean under augmentation and demonstrates that uniform weighting is not always MVUE, motivating non-uniform strategies. In the PAC setting, ERM can stall generalization when $\alpha$ exceeds $1/2$, but the authors present universal algorithms based on PU learning and XOR formulations that achieve vanishing generalization error for arbitrary VC dimension and contamination. Overall, the paper reveals fundamental limits of ERM in synthetic-data scenarios and provides principled algorithms to robustly learn under ongoing data contamination.

Abstract

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.

Theorems & Definitions (31)

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• Lemma 2