Uniform Race: Parameter-Free Approximate Rejection Sampling

2026-09-28

AI Summary

This study addresses the limitation of approximate rejection sampling, which relies on known distributional properties to set thresholds. To overcome this, we propose Uniform Race, a parameter-free algorithm that constructs a scoring mechanism by combining importance weights with uniform random variables. By selecting the candidate with the maximum score, the method achieves an exact approximation of the target distribution without requiring preset thresholds, simultaneously attaining the optimal error upper bound across all fixed thresholds and uniquely determining selection probabilities under specific conditions. Theoretical analysis demonstrates that Uniform Race yields an exponential reduction in total variation distance error compared to baseline methods. Furthermore, experiments on mathematical reasoning tasks with large language models empirically validate both the efficiency and accuracy of the proposed approach.

Abstract

We study approximate sampling: given $N$ independent samples from a proposal distribution $μ$, the goal is to select one whose distribution is close to a target $π$ 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 $(π,μ)$ 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 $π$. 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 $(π,μ)$ 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.

Problem

• approximate sampling
• rejection sampling
• parameter-free
• acceptance threshold
• importance weights

Innovation

• Approximate Rejection Sampling
• Parameter-Free Algorithm
• Importance Weights
• Total Variation Error
• Test-Time Scaling