The Fundamental Limits of Structure-Agnostic Functional Estimation
不依赖结构的泛函估计之基本极限AI 解读 ↓
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不依赖结构的泛函估计之基本极限
- 研究问题
- 当只对干扰函数估计施加高层次收敛率要求、保持黑箱兼容性时，一阶去偏方法是否已达到泛函估计的基本最优界限？
- 方法与思路
- 作者形式化使用黑箱干扰函数估计的结构无关泛函估计问题，推导极小极大下界，并比较一阶方法与依赖函数空间结构的高阶方法。
- 材料陈述的贡献
- 摘要称，在所定义的结构无关框架中一阶方法具有最优性，揭示黑箱兼容性与利用已知结构构造更优高阶估计之间的权衡。
- 阅读关注点
- 应核对结构无关框架允许的信息、风险与参数空间定义，以及下界适用的泛函和收敛率条件，明确最优性相对于哪些估计程序。
- 阅读边界
- 这是 arXiv 预印本，同行评审与期刊接收均未核实；摘要未列出完整下界条件，不能据此认为所有问题中的一阶方法均最优。
原文摘要
Many recent developments in causal inference, and functional estimation problems more generally, have been motivated by the fact that classical one-step (first-order) debiasing methods, or their more recent sample-split double machine-learning avatars, can outperform plugin estimators under surprisingly weak conditions. These first-order corrections improve on plugin estimators in a black-box fashion, and consequently are often used in conjunction with powerful off-the-shelf estimation methods. These first-order methods are however provably suboptimal in a minimax sense for functional estimation when the nuisance functions live in Holder-type function spaces. This suboptimality of first-order debiasing has motivated the development of "higher-order" debiasing methods. The resulting estimators are, in some cases, provably optimal over Holder-type spaces, but both the estimators which are minimax-optimal and their analyses are crucially tied to properties of the underlying function space. In this paper we investigate the fundamental limits of structure-agnostic functional estimation, where relatively weak conditions are placed on the underlying nuisance functions. We show that there is a strong sense in which existing first-order methods are optimal. We achieve this goal by providing a formalization of the problem of functional estimation with black-box nuisance function estimates, and deriving minimax lower bounds for this problem. Our results highlight some clear tradeoffs in functional estimation -- if we wish to remain agnostic to the underlying nuisance function spaces, impose only high-level rate conditions, and maintain compatibility with black-box nuisance estimators then first-order methods are optimal. When we have an understanding of the structure of the underlying nuisance functions then carefully constructed higher-order estimators can outperform first-order estimators.