Existence of anti-derivative loss functions for functional scores

Determine whether an anti-derivative loss function exists for a given functional score in the infinite-dimensional functional-score framework.

Background

The paper generalizes loss-function-based orthogonality to infinite-dimensional functional scores for estimating time-varying heterogeneous treatment effects. It notes that, unlike in finite-dimensional settings, a functional score need not universally correspond to the derivative of a loss function. The reverse construction—recovering a loss function from a given functional score—is therefore substantially more difficult, and the existence of such an anti-derivative loss function is left uncertain.