Inference-Time Alignment for Diffusion Models via Variationally Stable Doob's Matching

TL;DR

This paper introduces a new inference-time guidance method for diffusion models using Doob's $h$-transform, providing provable convergence guarantees and robustness to high-dimensional data.

Contribution

It proposes variationally stable Doob's matching, a novel guidance estimation framework with theoretical convergence guarantees and adaptability to low-dimensional structures.

Findings

Provides non-asymptotic convergence rates for guidance estimation.

Proves convergence guarantees for generated distributions in Wasserstein distance.

Demonstrates robustness to high-dimensional data under low-dimensional assumptions.

Abstract

Inference-time alignment for diffusion models aims to adapt a pre-trained reference diffusion model toward a target distribution without retraining the reference score network, thereby preserving the generative capacity of the reference model while enforcing desired properties at the inference time. A central mechanism for achieving such alignment is guidance, which modifies the sampling dynamics through an additional drift term. In this work, we introduce variationally stable Doob's matching, a novel framework for provable guidance estimation grounded in Doob's $h$-transform. Our approach formulates guidance as the gradient of logarithm of an underlying Doob's $h$-function and employs gradient-regularized regression to simultaneously estimate both the $h$-function and its gradient, resulting in a consistent estimator of the guidance. Theoretically, we establish non-asymptotic convergence rates for the estimated guidance. Moreover, we analyze the resulting controllable diffusion processes and prove non-asymptotic convergence guarantees for the generated distributions in the 2-Wasserstein distance. Finally, we show that variationally stable guidance estimators are adaptive to unknown low dimensionality, effectively mitigating the curse of dimensionality under low-dimensional subspace assumptions.