Constrained Diffusion from Binary Evaluations via Learned Doob h-Functions

Conditional generation under constraints that can only be checked on a finished sample, such as a molecular property filter, remains challenging for diffusion models: most guidance methods require differentiable measurement operators, paired conditional data, or rollouts of the base model at every step. We present a framework grounded in the Doob $h$-transform that reduces conditional generation to learning a single function, the probability that a noisy sample denoises into the constraint set. We learn this function from binary clean-sample evaluations alone, by squared-error regression on forward-noised labels; the base model is frozen, one lightweight head is trained per constraint, and noisy-state classifier guidance arises as a special case. The exact guidance step fades where the constraint signal provably vanishes and carries a factor of the inverse probability across samples, which normalized guidance discards: with the schedule and mean step size held fixed, equalizing the step costs 17.0 to 18.5 points of constraint satisfaction on CIFAR-10 at comparable FID, across independently trained estimators. Guidance is more than $50\times$ cheaper per satisfying sample than rejection sampling at a 1.7% base rate, raises a Lipinski-type property filter on QM9 from 54.3% to 96.8%, and lifts satisfaction on Stable Diffusion 1.5 from 7.2% to 60.0% at one sixteenth of the cost of derivative-free selection, which reaches 9.0%. These results demonstrate that binary evaluations of clean samples are sufficient supervision for constrained generation across pixel, latent, and molecular diffusion.