  The phenomenon of model collapse, introduced in (Shumailov et al., 2023),
refers to the deterioration in performance that occurs when new models are
trained on synthetic data generated from previously trained models. This
recursive training loop makes the tails of the original distribution disappear,
thereby making future-generation models forget about the initial (real)
distribution. With the aim of rigorously understanding model collapse in
language models, we consider in this paper a statistical model that allows us
to characterize the impact of various recursive training scenarios.
Specifically, we demonstrate that model collapse cannot be avoided when
training solely on synthetic data. However, when mixing both real and synthetic
data, we provide an estimate of a maximal amount of synthetic data below which
model collapse can eventually be avoided. Our theoretical conclusions are
further supported by empirical validations.
