  Discrete diffusion modeling is a promising framework for modeling and
generating data in discrete spaces. To sample from these models, different
strategies present trade-offs between computation and sample quality. A
predominant sampling strategy is predictor-corrector $\tau$-leaping, which
simulates the continuous time generative process with discretized predictor
steps and counteracts the accumulation of discretization error via corrector
steps. However, for absorbing state diffusion, an important class of discrete
diffusion models, the standard forward-backward corrector can be ineffective in
fixing such errors, resulting in subpar sample quality. To remedy this problem,
we propose a family of informed correctors that more reliably counteracts
discretization error by leveraging information learned by the model. For
further efficiency gains, we also propose $k$-Gillespie's, a sampling algorithm
that better utilizes each model evaluation, while still enjoying the speed and
flexibility of $\tau$-leaping. Across several real and synthetic datasets, we
show that $k$-Gillespie's with informed correctors reliably produces higher
quality samples at lower computational cost.
