Efficient and Unbiased Sampling from Boltzmann Distributions via Variance-Tuned Diffusion Models

Abstract

Score-based diffusion models (SBDMs) are powerful amortized samplers for Boltzmann distributions; however, imperfect score estimates bias downstream Monte Carlo estimates. Classical importance sampling (IS) can correct this bias, but computing exact likelihoods requires solving the probability-flow ordinary differential equation (PF–ODE), a procedure that is prohibitively costly and scales poorly with dimensionality. We introduce Variance-Tuned Diffusion Importance Sampling (VT-DIS), a lightweight post-training method that adapts the per-step noise covariance of a pretrained SBDM by minimizing the $\alpha$-divergence $(\alpha=2)$ between its forward diffusion and reverse denoising trajectories. VT-DIS assigns a single trajectory-wise importance weight to the joint forward–reverse process, yielding unbiased expectation estimates at test time with negligible inference-time overhead compared to standard sampling. On the DW-4, LJ-13, and alanine-dipeptide benchmarks, VT-DIS achieves effective sample sizes of approximately 80%, 35%, and 3.5%, respectively, while using only a fraction of the computational budget required by vanilla diffusion + IS or PF-ODE–based IS.