arXiv:2606.24074cs.CCcs.AI2026-06被引 2

提出量化验证随机模型可靠性的最小代价方法,为高精度验证提供理论依据。

Token Complexity of Certifying Stochastic-Oracle Reliability

  • 基于SPRT设计自适应验证框架,通过累积对数似然证据判断可靠性
  • 在误差可控条件下,给出认证所需的最低预期调用次数上界
  • 证明该方法在小误差下达到最优,适合需要严格可靠性验证的场景

本文针对随机预言机(stochastic oracle)的可靠性验证问题,引入认证令牌复杂度的概念:在控制双侧错误率的前提下,区分达到目标可靠性水平与低于阈值的预言机所需的最小期望调用成本。提出一种基于序贯概率比检验(SPRT)的认证型随机图灵机(SOTM),通过查询预言机、计算二元正确性评分,并在累积对数似然证据超过决策阈值时停止。该机制几乎必然终止,满足可靠性区域间的双侧误差保证,并给出认证令牌复杂度的显式上界,其依赖于可靠性阈值、误差上限和每轮平均调用成本。进一步建立匹配的信息论下界:即使采用自适应查询,所有有界误差的认证SOTM在误差趋于零时,其主导阶的期望调用成本与SPRT构造一致。二者共同刻画了小误差情形下的最优认证令牌复杂度。

原文摘要 · Abstract (English)

Wang~\cite{Wang2026} introduced the Stochastic-Oracle Turing Machine (SOTM) framework and defined token complexity as the minimum expected cost of interacting with a stochastic oracle needed to attain a specified solution quality for a task. This paper develops an analogous notion for certifying the reliability of a stochastic oracle on a given domain. Certification token complexity is the minimum expected token cost required, with controlled error probability, to distinguish oracles that meet a target reliability level from those that fall below a lower reliability threshold. We construct an SPRT-based certification SOTM that queries the oracle, computes binary correctness scores, and stops when the accumulated log-likelihood evidence crosses a decision threshold. The SOTM halts almost surely, satisfies the desired two-sided error guarantee over the reliability regions to be certified, and yields an explicit upper bound on certification token complexity in terms of the reliability thresholds, the error bound, and the expected per-turn token cost. We then establish a matching information-theoretic lower bound: even with adaptive queries, every error-bounded certification SOTM must incur the same leading-order expected token cost as the SPRT-based construction as the prescribed error bound tends to zero. Together, these bounds characterize the leading-order certification token complexity in the small-error regime.

可靠性验证随机预言机统计推断信息论

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