We study a constructive procedure that approximates Gateaux derivatives for statistical functionals by finite-differencing, with attention to causal inference functionals. We focus on the case where probability distributions are not known a priori but need also to be estimated from data, leading to empirical Gateaux derivatives, and study relationships between empirical, numerical, and analytical Gateaux derivatives. Starting with a case study of counterfactual mean estimation, we verify the exact 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 statistical benefits. We study more complicated functionals such as dynamic treatment regimes and the linear-programming formulation for policy optimization infinite-horizon Markov decision processes. In the case of the latter, this approach can be used to approximate bias adjustments in the presence of arbitrary constraints, illustrating the usefulness of constructive approaches for Gateaux derivatives. We find that, omitting unfavorable dimension dependence of smoothing, although rate-double robustness permits for coarser rates of perturbation size than implied by generic approximation analysis of finite-differences for the case of the counterfactual mean, this is not the case for the infinite-horizon MDP policy value.

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 setting 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.

We study a constructive procedure that approximates Gateaux derivatives for statistical functionals by finite-differencing, with attention to causal inference functionals. We focus on the case where probability distributions are not known a priori but need also to be estimated from data, leading to empirical Gateaux derivatives, and study relationships between empirical, numerical, and analytical Gateaux derivatives. Starting with a case study of counterfactual mean estimation, we verify the exact 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 statistical benefits. We study more complicated functionals such as dynamic treatment regimes and the linear-programming formulation for policy optimization infinite-horizon Markov decision processes.

We study a constructive procedure that approximates Gateaux derivatives for statistical functionals by finite-differencing, with attention to causal inference functionals. We focus on the case where probability distributions are not known a priori but need also to be estimated from data, leading to empirical Gateaux derivatives, and study relationships between empirical, numerical, and analytical Gateaux derivatives. Starting with a case study of counterfactual mean estimation, we verify the exact 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 statistical benefits. We study more complicated functionals such as dynamic treatment regimes and the linear-programming formulation for policy optimization infinite-horizon Markov decision processes.

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.