The era of proliferation of large language and image generation models begs the question of what happens if models are trained on the synthesized outputs of other models. 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 investigate this phenomenon within the context of high-dimensional regression with Gaussian data, considering both low- and high-dimensional asymptotics. We derive analytical formulas that quantitatively describe this phenomenon in both under-parameterized and over-parameterized regimes. We show how test error increases linearly in the number of model iterations in terms of all problem hyperparameters (covariance spectrum, regularization, label noise level, dataset size) and further isolate how model collapse affects both bias and variance terms in our setup.

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.