We study a constructive algorithm that approximates Gateaux derivatives for statistical functionals by finite differencing, with a focus on functionals that arise in causal inference. We study the case where probability distributions are not known a priori but need to be estimated from data. These estimated distributions lead to empirical Gateaux derivatives, and we study the relationships between empirical, numerical, and analytical Gateaux derivatives. Starting with a case study of the interventional mean (average potential outcome), we delineate the relationship between finite differences and the analytical Gateaux derivative. We then derive requirements on the rates of numerical approximation in perturbation and smoothing that preserve the statistical benefits of one-step adjustments, such as rate double robustness. We then study more complicated functionals such as dynamic treatment regimes, the linear-programming formulation for policy optimization in infinite-horizon Markov decision processes, and sensitivity analysis in causal inference. More broadly, we study optimization-based estimators, since this begets a class of estimands where identification via regression adjustment is straightforward but obtaining influence functions under minor variations thereof is not. The ability to approximate bias adjustments in the presence of arbitrary constraints illustrates the usefulness of constructive approaches for Gateaux derivatives. We also find that the statistical structure of the functional (rate double robustness) can permit less conservative rates for finite-difference approximation. This property, however, can be specific to particular functionals; e.g., it occurs for the average potential outcome (hence average treatment effect) but not the infinite-horizon MDP policy value.

翻译：我们研究了一种通过有限差分近似统计泛函的加托导数的构造性算法，重点关注因果推断中出现的泛函。我们研究概率分布并非先验已知而需从数据中估计的情形。这些估计分布引出了经验加托导数，我们探究了经验加托导数、数值加托导数与解析加托导数之间的关系。以干预均值（平均潜在结果）为案例，我们阐明了有限差分与解析加托导数之间的关联。随后，我们推导了扰动与平滑中数值近似速率需满足的条件，以保留单步调整（如速率双稳健性）的统计优势。接着，我们研究了更复杂的泛函，包括动态治疗方案、无限时域马尔可夫决策过程中用于策略优化的线性规划公式，以及因果推断中的敏感性分析。更广泛地，我们聚焦于基于优化的估计量，因为这类估计量对应着一类估计目标：通过回归调整进行识别较为直接，但获取其微小变体下的影响函数却非易事。在存在任意约束的条件下近似偏差调整的能力，凸显了加托导数构造性方法的实用性。我们还发现，泛函的统计结构（速率双稳健性）可允许对有限差分近似采用较保守性更小的速率。然而，该性质可能仅特定于某些泛函：例如，它适用于平均潜在结果（进而适用于平均处理效应），但不适用于无限时域MDP策略价值。