Training-Free Adaptation of Diffusion Models via Doob's $h$-Transform

Adaptation methods have been a workhorse for unlocking the transformative power of pre-trained diffusion models in diverse applications. Existing approaches often abstract adaptation objectives as a reward function and steer diffusion models to generate high-reward samples. However, these approaches can incur high computational overhead due to additional training, or rely on stringent assumptions on the reward such as differentiability. Moreover, despite their empirical success, theoretical justification and guarantees are seldom established. In this paper, we propose $\texttt{DOIT}$ ($\textbf{D}$oob-$\textbf{O}$riented $\textbf{I}$nference-time $\textbf{T}$ransformation), a training-free adaptation method for generic, non-differentiable rewards. We develop two variants of this method: $\texttt{DOIT-Proto}$, a full-rollout simulation-based algorithm and $\texttt{DOIT-Practical}$, an efficient surrogate algorithm. The key idea is a measure transport formulation that seeks to transport the pre-trained generative distribution to a high-reward target distribution. We leverage Doob's $h$-transform to realize this transport, which induces a dynamic correction to the diffusion sampling process and admits simulation-based computation without modifying the pre-trained model. Theoretically, we establish a high-probability convergence guarantee to the target high-reward distribution for $\texttt{DOIT-Proto}$ by characterizing the approximation error in the dynamic Doob correction. Empirically, $\texttt{DOIT-Practical}$ consistently outperforms state-of-the-art baselines on D4RL offline RL benchmarks while preserving sampling efficiency.