Diffusion Models Memorize in Training -- and Generalize in Inference

Diffusion models generalize well in practice. Paradoxically, an optimal diffusion model fully memorizes the training data and therefore fails to generalize, raising the question of what induces generalization in a real diffusion model. We show that diffusion models progressively overfit the denoising training objective, creating a generalization gap between validation and training performance at intermediate noise levels. In a fully analytic 2D toy model with a controlled denoising error, we trace this gap to the interaction between model error and the density of the data distribution's support. The optimal denoising flow field localizes sharply around individual training points, whereas model error suppresses exact recall of training points, yielding a smooth, generalizing flow field. Finally, we examine how training-time overfitting manifests along inference trajectories. We find that predictions made from intermediate trajectory states occupy a distinct feature-space regime from predictions made from noised training and validation images. As training progresses and model size increases, these predictions develop greater relative affinity to the training data, despite the absolute similarity to the validation data not decreasing. Together, these findings show that the denoising objective and inference trajectories express structural memorization differently.