Stratified Hazard Sampling: Minimal-Variance Event Scheduling for CTMC/DTMC Discrete Diffusion and Flow Models

Abstract

Uniform-noise discrete diffusion and flow models (e.g., D3PM, SEDD, UDLM, DFM) generate sequences non-autoregressively by iteratively refining randomly initialized vocabulary tokens through multiple context-dependent replacements. These models are typically formulated as time-inhomogeneous CTMC/DTMC processes and sampled using independent Bernoulli change decisions at each discretization step. This induces Poisson-binomial variance in per-position jump counts that grows with the number of required edits, leading to the characteristic under-editing (residual noise) and over-editing (cascading substitutions) failure modes that degrade sample quality, especially under tight discretization budgets. In contrast, absorbing-state (mask-start) models avoid this instability by allowing each position to jump at most once. We propose Stratified Hazard Sampling (SHS), a training-free, drop-in, and hyperparameter-free inference principle for any sampler that admits a stay-vs.-replace decomposition. SHS models per-token edits as events driven by cumulative hazard (CTMC) or cumulative jump mass (DTMC) and places events by stratifying this cumulative quantity: with a single random phase per position, a token is updated whenever its accumulated hazard crosses unit-spaced thresholds. This preserves the expected number of jumps while achieving the minimum possible conditional variance among unbiased integer estimators (bounded by 1/4 for any fixed cumulative mass), without altering per-jump destination sampling and thus retaining multimodality. Experiments on uniform-noise discrete diffusion language models show that SHS consistently improves sample quality. We further show that SHS improves robustness under token-level blacklist filtering, with benefits increasing as lexical constraints grow more severe.