Diffusion Models with Heavy-Tailed Targets: Score Estimation and Sampling Guarantees

Score-based diffusion models have emerged as a powerful framework for generative modeling, with score estimation as a central statistical bottleneck. However, existing theoretical guarantees largely focus on light-tailed distributions or impose restrictive assumptions such as compact support, limiting their applicability to heavy-tailed data encountered in practice. In this work, we study standard score-based diffusion models for heavy-tailed distributions in a Sobolev class with smoothness $\beta >0$, covering both exponential and polynomial tail decay characterized by a tail parameter $\gamma$. Based on kernel density estimation, we derive sharp minimax rates for score estimation, revealing a qualitative distinction between the two regimes: under exponential tails, the estimation rate matches the light-tailed benchmark up to polylogarithmic factors, whereas under polynomial tails it explicitly depends on the tail parameter $\gamma$. We further establish the convergence guarantees for the associated continuous-time reverse diffusion dynamics. For exponential tails and $\beta \in (0,2]$, the generated distribution converges in total variation at the rate $n^{-\beta /(2\beta +d)}$ up to logarithmic factors, matching the light-tailed benchmark. For polynomial tails, we obtain a $\gamma$-dependent total variation bound, while leaving its optimality open. These results characterize the statistical limits of score estimation and the resulting sampling accuracy for heavy-tailed targets, extending diffusion theory beyond the light-tailed setting.