StAD: Stein Amortized Divergence for Fast Likelihoods with Diffusion and Flow

Diffusion models achieve strong generation quality, diversity, and distribution coverage, but their performance often comes with expensive inference. In this work, we propose Stochastic Transition-Map Distillation (STMD), a teacher-free framework for accelerating diffusion model inference while preserving probabilistic sample generation. In contrast to score-based diffusion models, whose denoising parametrization models the mean of the posterior distribution, STMD distills the full transition map associated with the sampling stochastic differential equation (SDE). We parameterize these SDE transitions with a conditional Mean Flow model, yielding a one- or few-step stochastic sampler that retains the transition structure of the underlying diffusion process. This perspective is especially useful for downstream tasks that require stochastic inference, such as diffusion posterior sampling, inverse problems, and energy-based fine-tuning. Compared to recent distillation methods, STMD requires no pretrained teacher, bi-level optimization, or trajectory simulation and caching, enabling efficient and scalable training. We derive convergence bounds for our method in the Wasserstein distance, providing a strong theoretical foundation for our approach, and validate STMD on various image generation examples on the MNIST, CIFAR-10, and CelebA datasets.

A fundamental tension exists in the large-step inference of diffusion models via their deterministic probability flow ordinary differential equation (PF-ODE) trajectories, which we identify as the contractivity trap: efficient inference favors large step sizes, while aggressive steps and highly expressive denoisers can undermine contraction-based stability certificates for error suppression. To address this, we propose SteinDiff, a step-wise inference-time stabilization framework that employs Stein-derived corrections without requiring reference samples. Specifically, SteinDiff introduces a geometry-aware residual correction mechanism that regularizes large-step solver updates without retraining. To this end, we derive a closed-form Stein correction coefficient for step-wise solver adjustment, enabling reference-free adaptation to local data geometry. We further establish a score-controlled perturbation bound under distributional shifts and provide a complementary Stein perspective on EDM-style parameterizations. Extensive experiments demonstrate that SteinDiff mitigates severe artifacts and improves generative quality across large-step inference settings.

We study the problem of training diffusion and flow generative models to sample from target distributions defined by an exponential tilting of a base density; a formulation that subsumes both sampling from unnormalized densities and reward fine-tuning of pre-trained models. This problem can be approached from a stochastic optimal control (SOC) perspective, using adjoint-based or score matching methods, or from a non-equilibrium thermodynamics perspective. We provide a unified framework encompassing these approaches and make three main contributions: (i) bias-variance decompositions revealing that Adjoint Matching/Sampling and Novel Score Matching have finite gradient variance, while Target and Conditional Score Matching do not; (ii) norm bounds on the lean adjoint ODE that theoretically support the effectiveness of adjoint-based methods; and (iii) adaptations of the CMCD and NETS loss functions, along with novel Crooks and Jarzynski identities, to the exponential tilting setting. We validate our analysis with reward fine-tuning experiments on Stable Diffusion 1.5 and 3.