BARS: Budget-Aware Sequential Sampling with Refusal for Majority Identification under Unknown Distributions

Shenxi Liu, Yiwei Li, Mingyang Zhao, Yuhang Tian, Kan Li

Repeated sampling with majority selection is an effective means of enforcing self-consistency in large language models. Recently, a variant has also been introduced that terminates early to save costs when the output distribution is favorable. But it still cannot yield a correct confidence estimate from the answer distribution. For this issue, we formalize this as majority identification under an unknown distribution with a budget and a target error rate, and propose BARS, a sequential rule certifying the top-two gap and returning an early stop, a budgeted majority vote, or a refusal when feasibility cannot be established. A refusal is a statement about the rule and its budget, not a claim that a question is unanswerable, and it tells practitioners when to widen the budget or switch models. A certified mode carries proven bounds on the joint answering error and on refusing an instance the budget could handle, while a practical mode, used in our experiments, replaces them with empirical calibration. On synthetic instances drawn from known distributions over thousands of seeds, the difficulty-driven behavior is recovered and the price of the certified mode is measured.On GSM8K the…