Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach

2026-02-05

AI Summary

This work addresses the limitation of soft guidance in diffusion models, which cannot guarantee satisfaction of prescribed constraints, by proposing a hard-constrained conditional generation method that ensures generated samples satisfy specified events with probability one. The approach leverages Doob’s h-transform and martingale theory to construct an explicit drift correction mechanism that operates without modifying the pre-trained score network. It employs off-policy learning to estimate the conditioning function and its gradient from pre-trained trajectories. Innovatively, the method introduces martingale loss and martingale covariance loss, yielding the first non-asymptotic error bounds for hard-constrained diffusion guidance and rigorously characterizing the impact of score approximation and guidance estimation errors on sampling quality. Non-asymptotic convergence guarantees are established under both total variation and Wasserstein distances, and experiments demonstrate the method’s effectiveness in enforcing hard constraints and generating rare-event samples.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

We study conditional generation in diffusion models under hard constraints, where generated samples must satisfy prescribed events with probability one. Such constraints arise naturally in safety-critical applications and in rare-event simulation, where soft or reward-based guidance methods offer no guarantee of constraint satisfaction. Building on a probabilistic interpretation of diffusion models, we develop a principled conditional diffusion guidance framework based on Doob's h-transform, martingale representation and quadratic variation process. Specifically, the resulting guided dynamics augment a pretrained diffusion with an explicit drift correction involving the logarithmic gradient of a conditioning function, without modifying the pretrained score network. Leveraging martingale and quadratic-variation identities, we propose two novel off-policy learning algorithms based on a martingale loss and a martingale-covariation loss to estimate h and its gradient using only trajectories from the pretrained model. We provide non-asymptotic guarantees for the resulting conditional sampler in both total variation and Wasserstein distances, explicitly characterizing the impact of score approximation and guidance estimation errors. Numerical experiments demonstrate the effectiveness of the proposed methods in enforcing hard constraints and generating rare-event samples.

Problem

• conditional generation
• hard constraints
• diffusion models
• rare-event simulation
• safety-critical applications

Innovation

• conditional diffusion
• hard constraints
• Doob's h-transform
• martingale loss
• rare-event simulation