Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach

TL;DR

This paper introduces a probabilistic framework for guiding diffusion models under strict constraints, ensuring samples meet specific conditions with high reliability, especially in safety-critical and rare-event scenarios.

Contribution

It develops a principled conditional guidance method using Doob's h-transform and proposes two novel off-policy learning algorithms with theoretical guarantees.

Findings

Effective enforcement of hard constraints in diffusion models.

Successful generation of rare-event samples.

Theoretical bounds on sampling errors.

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. The code of the numerical experiments can be found at https://github.com/ZhengyiGuo2002/CDG_Finance.