Constructing asymptotically valid confidence intervals through a valid central limit theorem is crucial for A/B tests, where a classical goal is to statistically assert whether a treatment plan is significantly better than a control plan. In some emerging applications for online platforms, the treatment plan is not a single plan, but instead encompasses an infinite continuum of plans indexed by a continuous treatment parameter. As such, the experimenter not only needs to provide valid statistical inference, but also desires to effectively and adaptively find the optimal choice of value for the treatment parameter to use for the treatment plan. However, we find that classical optimization algorithms, despite of their fast convergence rates under convexity assumptions, do not come with a central limit theorem that can be used to construct asymptotically valid confidence intervals. We fix this issue by providing a new optimization algorithm that on one hand maintains the same fast convergence rate and on the other hand permits the establishment of a valid central limit theorem. We discuss practical implementations of the proposed algorithm and conduct numerical experiments to illustrate the theoretical findings.
翻译:通过一个有效的中央限值理论来构建无效的有效信任间隔对于A/B测试至关重要,在A/B测试中,古典目标是从统计上确定治疗计划是否比控制计划好得多。在一些新出现的在线平台应用中,治疗计划不是一个单一的计划,而是包含由连续处理参数索引的无限的连续计划。因此,实验者不仅需要提供有效的统计推断,而且还希望有效和适应地找到治疗参数用于治疗计划的最佳价值选择。然而,我们发现,传统优化算法尽管在融合假设下快速趋同率,但并不带有核心限制的标值,而可用于构建尽可能有效的信任间隔。我们通过提供一种新的优化算法来解决这个问题,一方面保持同样的快速趋同率,另一方面允许建立有效的中央限值。我们讨论拟议的算法的实际执行情况,并进行数字实验,以说明理论结论。