Limiting distribution under recursive training of diffusion models

Determine whether the sequence of model distributions (\hat p^i) produced by recursively training a score-based diffusion model on mixed data q_i = \alpha\,data + (1-\alpha)\,\hat p^i converges to a limiting distribution as i \to \infty, and, if convergence occurs, characterize how the limiting distribution depends on the fresh–data proportion \alpha and on the true data distribution data.

Background

The paper analyzes recursive training of score-based diffusion models where, at each generation i, the training distribution is the mixture q_i = \alpha\,data + (1-\alpha)\,\hat p^i of fresh samples from the true data distribution and synthetic samples from the current model. It establishes lower and upper bounds on intra-generation and accumulated divergences, showing how fresh data mitigates error accumulation and how score estimation errors propagate.

While the analysis provides finite-horizon bounds and discounted accumulation laws under small error regimes, it leaves open the asymptotic behavior of the recursion. Specifically, the existence and characterization of a limiting distribution for (\hat p^i) under recursive training are not resolved, motivating a precise convergence question tied to \alpha and data.