Model Collapse Demystified: The Case of Regression

In the era of large language models like ChatGPT, the phenomenon of "model collapse" refers to the situation whereby as a model is trained recursively on data generated from previous generations of itself over time, its performance degrades until the model eventually becomes completely useless, i.e the model collapses. In this work, we study this phenomenon in the simplified setting of kernel regression and obtain results which show a clear crossover between where the model can cope with fake data, and a regime where the model's performance completely collapses. Under polynomial decaying spectral and source conditions, we obtain modified scaling laws which exhibit new crossover phenomena from fast to slow rates. We also propose a simple strategy based on adaptive regularization to mitigate model collapse. Our theoretical results are validated with experiments.