AI 中文总结
本文针对数据驱动优化，提出EO+框架，证明无有效辅助信息时其仅能实现二阶改进，利用几何有效辅助信息并选择合适超参数可实现一阶改进，还构建了最大化一阶增益的方法并关联控制 variate 原理。
AI 中文摘要
近期数据与优化的融合催生了一系列旨在提升数据驱动优化决策统计性能的方法。然而，尽管许多这类方法直观上源于鲁棒性或正则化视角，其统计收益却往往不明确，即便存在收益，也仅能在特定案例中得到验证。本文采用“方向扰动”经验优化（Empirical Optimization, EO）视角，对数据驱动优化的公式进行系统剖析。我们将这类公式统称为“EO+”，其涵盖了正则化、分布鲁棒优化、迁移学习以及上下文优化的类似方法等诸多现有数据驱动优化方法。一方面，我们论证：若缺乏额外的、正确指定的辅助信息，任何EO+方法最多只能实现二阶改进，这为EO+的统计功效提供了“不存在免费午餐”的否定结论。另一方面，我们表明：当利用几何上有效的辅助信息时，通过选择远大于文献通常建议的超参数，可实现一阶改进。此外，我们基于超额风险估计构建了一种原则性方法，可通过系统知识或自助重采样来最大化一阶增益。我们还展示了该增益如何与控制 variate 原理（蒙特卡洛模拟文献中的一种方差缩减技术）相联系，这有助于解释为何几何上有效的辅助信息是必要的。
英文摘要
Recent proliferation of data-optimization integration has led to a range of methods that aim to improve the statistical performance of data-driven optimization decisions. However, while many of these methods are motivated intuitively from a robustness or regularization perspective, their resulting statistical benefits are often unclear and, even if available, are established on a case-by-case basis. We provide a systematic dissection of data-driven optimization formulations using the view of "directionally perturbed" empirical optimization (EO). Specifically, this umbrella of formulations, which we call "EO+", covers many existing data-driven optimization methods, including regularization, distributionally robust optimization, transfer learning, and analogous methods for contextual optimization. On the one hand, we argue that without additional, correctly specified, side information, any EO+ method can result in at most second-order improvements. This provides a negative conclusion, namely ``no free lunch is possible", on the statistical power of EO+. On the other hand, we show that when leveraging side information that is geometrically effective, achieving first-order improvements is possible by choosing hyperparameters that are significantly larger than what is typically suggested in the literature. Moreover, we construct a principled methodology based on excess risk estimation, via either system knowledge or bootstrap resampling, to maximize the first-order gain. We demonstrate how this gain connects to the control-variate principle, a variance reduction technique in the Monte Carlo simulation literature, which helps explain why geometrically effective side information is necessary.
