Training-Free Guided Diffusion for Planning: A Unified Framework via Doob’s h-Transform with Safety Guarantees

This paper studies the theoretical foundations of guidance mechanisms in continuous-time score-based diffusion models. We adopt Doob’s h-transform as a principled framework for characterizing ideal guided diffusion processes and analyze the discrepancy between ideal and approximate guidance. Our analysis provides explicit error bounds and yields probabilistic guarantees on satisfying prescribed constraints, which are particularly important for safety-critical planning. We further show that the Doob-based formulation induces a stochastic optimal control problem, enabling practical guidance design without additional model training. We demonstrate the effectiveness of the proposed framework on robotic navigation tasks, including language-conditioned planning.

Diffusion models are increasingly being used for robot planning, where a robot must generate trajectories while accounting for safety requirements and obstacles. A common approach is to guide the sampling process toward desirable trajectories, but it remains unclear when such guidance works and how it should be designed. In this work, we study guided diffusion-based planning using mathematical tools that describe how random paths can be steered toward desired outcomes. Our framework provides a principled way to design guidance with theoretical guarantees. The results establish a foundation for safer and more constraint-aware planning with diffusion models, illustrated through robot planning examples.