Improved Analysis of Score-based Generative Modeling: User-Friendly Bounds under Minimal Smoothness Assumptions
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In this paper, we focus on the theoretical analysis of diffusion-based generative modeling. Under an L^2-accurate score estimator, we provide convergence guarantees with polynomial complexity for any data distribution with second-order moment, by either employing an early stopping technique or assuming smoothness condition on the score function of the data distribution. Our result does not rely on any log-concavity or functional inequality assumption and has a logarithmic dependence on the smoothness. In particular, we show that under only a finite second moment condition, approximating the following in KL divergence in ϵ-accuracy can be done in Õ(d^2 log^2 (1/δ)/ϵ^2) steps: 1) the variance-δ Gaussian perturbation of any data distribution; 2) data distributions with 1/δ-smooth score functions. Our theoretical analysis also provides quantitative comparison between different discrete approximations and may guide the choice of discretization points in practice.
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