This paper proposes a statistically optimal approach for learning a function value using a confidence interval in a wide range of models, including general non-parametric estimation of an expected loss described as a stochastic programming problem or various SDE models. More precisely, we develop a systematic construction of highly accurate confidence intervals by using a moderate deviation principle-based approach. It is shown that the proposed confidence intervals are statistically optimal in the sense that they satisfy criteria regarding exponential accuracy, minimality, consistency, mischaracterization probability, and eventual uniformly most accurate (UMA) property. The confidence intervals suggested by this approach are expressed as solutions to robust optimization problems, where the uncertainty is expressed via the underlying moderate deviation rate function induced by the data-generating process. We demonstrate that for many models these optimization problems admit tractable reformulations as finite convex programs even when they are infinite-dimensional.

翻译：本文提出了一种统计最优方法，用于在广泛模型（包括描述为随机规划问题的期望损失的一般非参数估计或各种SDE模型）中利用置信区间学习函数值。更精确地说，我们通过采用基于中偏差原理的方法，系统性地构建了高精度置信区间。研究表明，所提出的置信区间在指数精度、极小性、一致性、误表征概率以及最终一致最精确（UMA）性质等准则意义上具有统计最优性。该方法建议的置信区间可表示为鲁棒优化问题的解，其中不确定性通过数据生成过程诱导的潜在中偏差率函数来刻画。我们证明，对于许多模型，即使这些优化问题具有无限维结构，它们仍可转化为有限凸规划的可处理形式。