  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.
