The paper introduces Uniform Race (UR), a parameter‑free algorithm for approximate rejection sampling that selects a sample by maximizing the ratio of its importance weight to an independent uniform random variable. UR achieves the same total variation error bound as the optimal rejection sampler that requires knowledge of the best acceptance threshold, and it outperforms both the budget‑calibrated rejection sampler and sampling importance resampling in certain settings. The authors also provide theoretical conditions under which UR’s selection probabilities are uniquely optimal and validate the method on large‑language‑model reasoning tasks.
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