From clinical development of cancer therapies to investigations into partisan bias, adaptive sequential designs have become increasingly popular method for causal inference, as they offer the possibility of improved precision over their non-adaptive counterparts. However, even in simple settings (e.g. two treatments) the extent to which adaptive designs can improve precision is not sufficiently well understood. In this work, we study the problem of Adaptive Neyman Allocation in a design-based potential outcomes framework, where the experimenter seeks to construct an adaptive design which is nearly as efficient as the optimal (but infeasible) non-adaptive Neyman design, which has access to all potential outcomes. Motivated by connections to online optimization, we propose Neyman Ratio and Neyman Regret as two (equivalent) performance measures of adaptive designs for this problem. We present Clip-OGD, an adaptive design which achieves $\widetilde{O}(\sqrt{T})$ expected Neyman regret and thereby recovers the optimal Neyman variance in large samples. Finally, we construct a conservative variance estimator which facilitates the development of asymptotically valid confidence intervals. To complement our theoretical results, we conduct simulations using data from a microeconomic experiment.

翻译：从癌症疗法的临床开发到对党派偏见的调查研究，自适应序贯设计已成为因果推断中越来越流行的方法，因为它们相比非自适应设计有可能提高精度。然而，即使在简单设置（例如两种处理）中，自适应设计能在多大程度上提高精度尚未得到充分理解。在本工作中，我们基于设计的潜在结果框架研究自适应奈曼分配问题，其中实验者试图构建一种自适应设计，使其几乎与最优（但不可行）的非自适应奈曼设计一样高效，后者能够获取所有潜在结果。受在线优化关联的启发，我们提出奈曼比率和奈曼遗憾作为该问题中自适应设计的两种（等价的）性能度量。我们提出Clip-OGD，一种自适应设计，它实现了$\widetilde{O}(\sqrt{T})$的期望奈曼遗憾，从而在大样本中恢复最优奈曼方差。最后，我们构建了一个保守方差估计量，以促进渐近有效置信区间的开发。为补充我们的理论结果，我们使用微观经济实验的数据进行模拟。