Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach

Researchers present a novel framework for conditional diffusion models that enforces hard constraints on generated samples using Doob's h-transform and martingale theory. The method enables safety-critical applications and rare-event simulation without requiring modifications to pretrained models, with theoretical guarantees on constraint satisfaction.

This research addresses a fundamental challenge in generative AI: ensuring that sampled outputs satisfy strict, hard constraints rather than soft probabilistic guidance. Traditional diffusion model guidance methods rely on reward signals or soft constraints that cannot guarantee constraint satisfaction—a critical limitation for safety-sensitive domains like finance, autonomous systems, and scientific simulation. The authors develop a mathematically rigorous framework grounded in stochastic analysis, leveraging Doob's h-transform to condition diffusion processes on prescribed events occurring with probability one.

The approach builds on the probabilistic interpretation of diffusion models by introducing explicit drift corrections based on the logarithmic gradient of a conditioning function. Crucially, this method preserves pretrained score networks without modification, reducing computational overhead and enabling application to existing models. The authors propose two novel off-policy learning algorithms using martingale and martingale-covariation losses, allowing practitioners to estimate guidance using only trajectories from the original model.

For the AI and finance sectors, this work offers significant practical implications. In quantitative finance, hard constraints ensure generated scenarios respect regulatory limits, risk bounds, or physical feasibility. In machine learning safety, guaranteed constraint satisfaction addresses alignment concerns. The non-asymptotic theoretical guarantees explicitly characterize errors from score approximation and guidance estimation, providing confidence intervals for practitioners.

Future development hinges on computational efficiency at scale and extension to high-dimensional problems. The publicly available code implementation suggests the authors anticipate adoption in financial modeling and scientific computing workflows.

• →Hard-constraint diffusion guidance uses Doob's h-transform to guarantee constraint satisfaction with probability one, solving limitations of soft guidance methods.
• →The framework works with pretrained diffusion models without modification, reducing computational costs and enabling broad applicability.
• →Two novel off-policy learning algorithms estimate conditioning functions using only trajectories from the original model.
• →Non-asymptotic theoretical guarantees in total variation and Wasserstein distances provide explicit error characterization for practitioners.
• →Applications in finance and safety-critical systems benefit from mathematically-principled constraint enforcement in generative modeling.