Data-Driven Influence Functions for Optimization-Based Causal Inference
This work addresses the challenge of deriving influence functions for optimization-based estimators in causal inference, which is incremental as it builds on existing methods for bias adjustments.
The paper tackles the problem of approximating Gateaux derivatives for causal inference functionals when distributions are estimated from data, showing that finite-differencing can preserve statistical benefits like rate double robustness for some functionals, such as the average treatment effect, but not for others like infinite-horizon MDP policy values.
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.