AI 中文总结
研究序贯抽样规则，通过自适应调整概率实现经验分布快速收敛到目标分布且保持样本不可预测。自平衡采样器有随机镜像下降解释，收敛速率快于标准，是熵正则化问题唯一解，弱偏差下计数过程收敛到特定过程，支持实用框架。
AI 中文摘要
许多序贯抽样实例，如审计和检查调度、代表性抽样及治疗分配，要求样本分布均匀且难以预测或利用。我们研究了一类序贯抽样规则，其自适应调整抽样概率，以更快地使经验分布收敛到目标分布，同时保持样本尽可能不可预测。所得自平衡采样器易于实现，在一类具有特定不变性的马尔可夫采样器中自然出现，并具有随机镜像下降解释。主要结果表明：（i）该自平衡采样器以最快的\(O(n^{-1})\)速率收敛，明确依赖于偏差参数，优于独立同分布采样的标准\(O(n^{-1/2})\)速率；（ii）它是一个自然熵正则化优化问题的唯一解，平衡了经验分布的收敛速率和样本的不可预测性；（iii）在弱偏差情况下，适当中心化的计数过程在扩散极限下收敛到奥恩斯坦 - 乌伦贝克过程。这些结果共同支持了一个实用框架，可减少重复选择和覆盖范围中的长间隙，同时不会使未来选择过于可预测。
英文摘要
Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.
Comments53 pages, 5 figures
