Non-asymptotic Convergence of Discrete-time Diffusion Models: New Approach and Improved Rate

The denoising diffusion model has recently emerged as a powerful generative technique that converts noise into data. While there are many studies providing theoretical guarantees for diffusion processes based on discretized stochastic differential equation (D-SDE), many generative samplers in real applications directly employ a discrete-time (DT) diffusion process. However, there are very few studies analyzing these DT processes, e.g., convergence for DT diffusion processes has been obtained only for distributions with bounded support. In this paper, we establish the convergence guarantee for substantially larger classes of distributions under DT diffusion processes and further improve the convergence rate for distributions with bounded support. In particular, we first establish the convergence rates for both smooth and general (possibly non-smooth) distributions having a finite second moment. We then specialize our results to a number of interesting classes of distributions with explicit parameter dependencies, including distributions with Lipschitz scores, Gaussian mixture distributions, and any distributions with early-stopping. We further propose a novel accelerated sampler and show that it improves the convergence rates of the corresponding regular sampler by orders of magnitude with respect to all system parameters. Our study features a novel analytical technique that constructs a tilting factor representation of the convergence error and exploits Tweedie's formula for handling Taylor expansion power terms.