Seasoning Generative Models for a Generalization Aftertaste

Despite the empirical success of score-based diffusion models, a complete theoretical understanding of how finite-sample learning, network parameterization, and numerical discretization jointly dictate generative quality remains underdeveloped. Existing sampling analyses often evaluate the generative performance conditional on an oracle score or a pre-specified error threshold. In this work, we establish a unified convergence and generalization framework for score-based diffusion models parameterized by practical ResNet-type architectures. We analyze the generalization and convergence properties from the practical finite-sample, discrete-time learning problem of the score function to the ideal continuous-time, population-level objective. Based on the generalization result of the learning problem of score function, we analyze the sampling process induced by the learned score function and provide an end-to-end total variation distance estimate for the generated terminal distribution. This estimate explicitly decomposes the overall generative error into four interpretable components: the truncation error of the forward process, the reverse-time discretization error, the generalization error incorporating both finite data and forward-time discretization, and the training optimization gap. Our results quantitatively characterize how the training sample size, temporal discretization grids, and optimization accuracy jointly control the final fidelity of samples generated by diffusion models.

Benign overfitting and double descent have come to shape our understanding of generalization in deep learning, establishing that overfitting is not only compatible with good generalization but can actively benefit it. Diffusion models share much of the machinery of standard deep learning, so it is natural to assume that they also exhibit these properties. In this work, we show that this assumption is largely incorrect. We first establish fundamental impossibility results showing that, unless the sample size grows exponentially with the data dimension, overfitting and good generalization cannot occur simultaneously. Consequently, the population loss follows a classical U-shaped curve in model complexity rather than exhibiting double descent. Analyzing a simplified setting, we identify a key difference between regression and score matching: regression benefits from an alignment between the target and the empirical covariance; score matching admits no such alignment, leaving overfitting irreparably harmful. We further identify implicit regularization stemming from time-smoothness of the score and early stopping during training as mechanisms that prevent such overfitting and verify our findings with high-dimensional image generation experiments. Our results reveal that generalization in diffusion models is governed by mechanisms distinct from those of traditional regression, motivating the development of new theory.

Diffusion models often generate novel samples even when the learned score is only coarse -- a phenomenon not accounted for by the standard view of diffusion training as density estimation. In this paper, we show that, under the manifold hypothesis, this behavior can instead be explained by coarse scores capturing the geometry of the data while discarding the fine-scale distributional structure of the population measure~$μ_{\scriptscriptstyle\mathrm{data}}$. Concretely, whereas estimating the full data distribution $μ_{\scriptscriptstyle\mathrm{data}}$ supported on a $k$-dimensional manifold is known to require the classical minimax rate $\tilde{\mathcal{O}}(N^{-1/k})$, we prove that diffusion models trained with coarse scores can exploit the regularity of the manifold support and attain a near-parametric rate toward a different target distribution. This target distribution has density uniformly comparable to that of~$μ_{\scriptscriptstyle\mathrm{data}}$ throughout any $\tilde{\mathcal{O}}\bigl(N^{-β/(4k)}\bigr)$-neighborhood of the manifold, where $β$ denotes the manifold regularity. Our guarantees therefore depend only on the smoothness of the underlying support, and are especially favorable when the data density itself is irregular, for instance non-differentiable. In particular, when the manifold is sufficiently smooth, we obtain that generalization -- formalized as the ability to generate novel, high-fidelity samples -- occurs at a statistical rate strictly faster than that required to estimate the full population distribution~$μ_{\scriptscriptstyle\mathrm{data}}$.