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的方法，用于在广泛模型中通过置信区间学习函数值，包括随机规划问题描述下的期望损失的一般非参数估计或各类随机微分方程模型。具体而言，我们通过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、最小性、一致性、误分类概率以及最终一致最准确性质方面满足标准，因此具有统计最优性。该方法建议的置信区间可表示为鲁棒优化问题的解，其中不确定性由数据生成过程诱导的底层中偏差速率函数所刻画。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问题来处理。