Kernel von Mises Formula of the Influence Function
Yaroslav Mukhin
The influence function (also known as the first variation and Fisher-Rao gradient)of a statistical functional is the Riesz representer of its derivative operator.It is the key analytic object in both the theory and implementation ofestimators in semiparametric statistical and machine learning models,e.g., one-step estimator, targeted learning, debiased machine learning.It is also essential for inference about their statistical, robustness andinterpretability properties,e.g., for finding confidence intervals, partial identification andmisspecification bounds, adversarial perturbations, influential data points.However, the analytic derivation of the influence function is often anobstruction to the broader adoption of these methods by practitioners.Toward automating this task,we derive a regularized representation of the influence function usingspectral theory of positive semidefinite kernels.Based on this representation we construct an estimator that:(i) is a nonparametric functional RKHS estimator;(ii) admits theoretical guarantees in function norms relevant for downstream tasks;(iii) can be computed via automatic differentiation or finite differences,without requiring analytic derivation by the user.