Uniform Race: Parameter-Free Approximate Rejection Sampling

Abstract:We study approximate sampling: given $N$ independent samples from a proposal distribution $\mu$, the goal is to select one whose distribution is close to a target $\pi$ specified only up to a normalizing constant. Block and Polyanskiy (2023) provide finite budget error bounds for approximate rejection sampling (RS) as a function of the acceptance threshold $M$. The threshold $M$ giving the smallest bound, however, depends on properties of $(\pi,\mu)$ that are typically unavailable from the observed sample. This raises a natural question: Can one attain the best RS guarantee without taking $M$ as input? We answer affirmatively by proposing a parameter-free sampling algorithm called uniform race (UR), based on importance weights, which are ratios of target to proposal probabilities (or densities). It divides each observed weight by an independent uniform random variable to form a score and returns the candidate with the largest score. For every budget $N$, its total variation error satisfies the RS upper bound for every fixed threshold $M$ simultaneously, thereby achieving the best such bound in hindsight. We also characterize its output distribution conditional on the largest score, identifying when it is exactly the target $\pi$. Uniform race has no larger total variation error than a natural budget-calibrated RS derived from Rohatgi et al. (2025) and sampling importance resampling (SIR). In particular, we exhibit instances where UR's error is exponentially smaller in $N$ than that of either baseline. Furthermore, we establish conditions under which attaining this RS guarantee for every $(\pi,\mu)$ uniquely determines the selection probabilities as those of UR. Finally, test-time scaling experiments on LLM math-reasoning tasks corroborate the theoretical comparisons and demonstrate that UR remains competitive in ground-truth accuracy without requiring threshold selection.

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2609.34639 [cs.LG]

(or arXiv:2609.34639v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.34639 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.