Convergence of Few-Step Importance Weights: A Cheap, Reference-Free Certificate

Diffusion and flow models increasingly use few solver steps as proposals for importance sampling, where the model likelihood enters the final estimator and numerical convergence becomes part of the sampling problem. Few-step likelihoods are only approximate, and confirming their accuracy would require exactly the expensive many-step reference that few-step sampling was meant to avoid. The standard cheap diagnostic, effective sample size, cannot fill this gap: it cancels a constant likelihood shift exactly, and its first-order response to a fluctuating error is governed by a squared-weight alignment that differs from the observable covariance governing estimation bias, so an observable can be biased while ESS is unchanged. We introduce a cheap, reference-free, a-posteriori certificate for the coupled log-weight discretization error. For a fixed source draw the log weight admits a structured expansion in the step size, so extrapolation from a few additional short solves reconstructs the error without invoking the reference; the certificate is provably sound once the step size falls below an explicit threshold, tightens as the budget grows, and uses no hand-tuned constants. The same expansion yields a same-sample truncation diagnostic, and an elementary bridge lemma converts per-sample error bars into a bound on the self-normalized estimate. Across controlled synthetic flows, one-step average-velocity maps, and energy-based molecular samplers, the certificate attains high empirical coverage, substantially reduces importance-sampling error on a real average-velocity map, and reaches near-reference-quality estimates at a fraction of the reference cost on a molecular system, whereas ESS-based budget rules accept severely biased budgets and cannot rank candidates at the degeneracy floor.