What Does a Discrete Diffusion Model Learn?

2026-07-06

AI Summary

This work clarifies the fundamental learning objective of discrete diffusion models—whether they function as denoisers, score ratio estimators, or bridge predictors—and systematically analyzes how different parameterizations affect training and sampling. By deriving the continuous-time Markov chain evidence lower bound (ELBO) for arbitrary noising processes, the authors establish a unified theoretical framework. They introduce the “Oracle Distance” theorem, proving that the negative ELBO decomposes into the sum of data entropy and the KL divergence along the forward path, and show that all noising processes share the same optimal value. This framework enables exact numerical verification of theoretical identities without approximation, accurately calibrates the initial ELBO value, and provides a cohesive interpretation of the relationships and distinctions among methods such as MDM, UDM, SEDD, and GIDD.

Abstract

What does a discrete diffusion model learn: a denoiser, a score ratio, or a bridge plug-in predictor? At the level of jump rates, these are one object in different coordinates, and reading a neural network in the wrong coordinate changes the process being trained and sampled. Starting with a rigorous derivation of the continuous-time Markov chain (CTMC) ELBO for any noising process, boundary terms included, we prove the \emph{Oracle Distance} theorem: the negative ELBO is exactly equal to the data entropy plus the path KL from the oracle reverse process to the learned one, not merely a bound. Its unique optimizer is therefore the conditional expectation of the true reverse jump rate given the current noisy state, and its irreducible cost is the rate at which the forward process $Z_t$ destroys information about the clean data $Z_0$, $-\tfrac{d}{dt}I(Z_0; Z_t)$, so every noising process shares the same best achievable negative ELBO: the data entropy. For sequences with token-factorizing noise, the oracle projection yields three exact coordinates for the optimizer: denoiser, cavity (bridge plug-in), and score, with closed-form conversions among them. This framework identifies which law each loss in the literature actually optimizes, recovering MDM, UDM, SEDD, and GIDD as special cases; explains why denoiser and cavity coincide for masked diffusion but not for uniform diffusion; proves that a denoiser parameterization makes the uniform ELBO diverge at initialization while the bridge plug-in stays finite; and calibrates ELBO implementations exactly at initialization. Every identity is verified numerically, without approximation, on an exactly solvable model.

Problem

• discrete diffusion model
• denoiser
• score ratio
• bridge plug-in predictor
• ELBO

Innovation

• discrete diffusion model
• continuous-time Markov chain
• ELBO
• oracle reverse process
• bridge plug-in predictor