What Does a Discrete Diffusion Model Learn?
2026-07-06 • Machine Learning
Machine LearningArtificial IntelligenceComputation and LanguageInformation Theory
AI summaryⓘ
The authors study what discrete diffusion models actually learn, showing that different ways to describe the model’s behavior (like denoising or scoring) are just different views of the same thing. They prove a key theorem linking the model’s training objective exactly to the information lost during the noising process. This means all noise processes share the same best achievable training score, related to the data’s inherent uncertainty. They also clarify how various existing model losses fit into this framework and explain practical implications for model parameterization and evaluation. All their results are confirmed precisely using a solvable example.
Discrete diffusion modelMarkov chainELBO (Evidence Lower Bound)Jump ratesOracle Distance theoremMutual informationDenoiserScore functionBridge plug-in predictor
Authors
Rodrigo Casado Noguerales, Bernhard Schölkopf, Thomas Hofmann, Aran Raoufi
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.