On the Edge of Memorization in Diffusion Models

When do diffusion models reproduce their training data, and when are they able to generate samples beyond it? A practically relevant theoretical understanding of this interplay between memorization and generalization may significantly impact real-world deployments of diffusion models with respect to issues such as copyright infringement and data privacy. In this paper, we propose a scientific and mathematical “laboratory” for investigating memorization and generalization in diffusion models trained on fully synthetic or natural image-like structured data. Within this setting, we theoretically characterize a crossover point wherein the weighted training loss of a fully generalizing model becomes greater than that of an underparameterized memorizing model at a critical value of model (under)parameterization. We then demonstrate via carefully-designed experiments that the location of this crossover predicts a phase transition in diffusion models trained via gradient descent, as our theory enables us to analytically predict the model size at which memorization becomes predominant. Our work provides an analytically tractable and practically meaningful setting for future theoretical and empirical investigations. Code for our experiments is available at https://github.com/DruvPai/diffusionmemgen.