Appendix: Budgeted Heterogeneous Treatment Effect Estimation

Proposition 1 (Shalit et al. (2017)). Let Φ : X → Z be a one-to-one representation function. Let f : Z × {0, 1} → Y be a hypothesis. Let G be a family of functions g : Z → Y . Assume that there exists a constant CΦ > 0, such that for fixed t ∈ {0, 1}, 1 CΦ ·`f,Φ(x, t) ∈ G, where `f,Φ(x, t) , ∫ Y l (f(Φ(x), t), y) p (y | x, t) dy. Let h(x) = f(Φ(x), 1)−f(Φ(x), 0) be an estimate for HTE. Assume l : Y × Y → R+ is the `2 loss, we have PEHE(h) ≤ 2 ( 1(f,Φ) + 0(f,Φ)) + 2CΦ · IPMG ( p1 , p Φ 0 ) − CY ,