Navigating Sparse Molecular Data with Stein Diffusion Guidance

Abstract: Stochastic optimal control (SOC) has recently emerged as a principled framework for fine-tuning diffusion models. However, its dependence on computationally intensive simulations makes it impractical for fast sampling. In parallel, a class of training-free approaches has been developed that guides diffusion models using off-the-shelf classifiers on predicted clean samples, bypassing the need to train classifiers on noisy data. These methods can be interpreted as approximate SOC schemes, using Tweedie's formula to estimate diffusion posteriors. In practice, however, such direct approximations can introduce significant errors, leading to unreliable guidance. In this work, we unify the strengths of both paradigms by proposing a novel training-free diffusion guidance framework based on a surrogate stochastic optimal control objective. We derive a new theoretical bound on the value function that reveals the necessity of correcting the approximate posteriors to remain faithful to the true diffusion posterior. To this end, we connect the problem with Stein variational inference, which seeks the steepest descent direction that minimizes the Kullback-Leibler discrepancy between the two posteriors. Our method, which we refer to as Stein Diffusion Guidance (SDG), introduces a principled correction mechanism and incorporates a novel running cost functional to enable effective guidance in low-density regions. Experiments on challenging molecular generation tasks demonstrate that SDG significantly outperforms standard training-free guidance methods, highlighting its potential for broader applications.