from the nonparametric model of distributions P ′ satisfying the same constraints as P to the real line. This functional is pathwise differentiable at P relative to the maximal tangent space with an efficient influence function f$_{π}$ given by By the bound on Y and the strong positivity assumption, there exists a universal constant M > 0 such that P (sup$_{π}$$_{∈}$$_{Π}$ | f$_{π}$ ( O ) | ≤ M ) = 1. We refer the interested reader to [14, Section 25.3] and [7] for definitions and proofs of these facts. Although the class will play a central role in the rest of this article, it is not necessary to master the derivation or properties of efficient influence functions to appreciate the contributions of this work. Suppose that we observe mutually independent data-structures O$_{1}$ ≡ ( X$_{1}$, A$_{1}$, Y$_{1}$ ) , . . . , O$_{n}$ ≡ ( X$_{n}$, A$_{n}$, Y$_{n}$ ) drawn from P . Let P$_{n}$ ≡ n − 1 ∑ n i $_{=1}$Dirac( O$_{i}$ ) denote the empirical measure. For every function f mapping an observation to the real line, we set Pf ≡ E [ f ( X, A, Y )] and P$_{n}$f ≡ n − 1 ∑ n i $_{=1}$f ( O$_{i}$ ). Our theorem will attain fast rates under the assumption that one has available an esti- Our theorem will attain fast rates under the assumption that one has available an esti- mator { ̂ V ( π ) : π ∈ Π } of {V ( π ) : π ∈ Π } that makes the following term small: Rem$_{n}$ ≡ sup π ∈ Π ∣ ̂ V ( π ) − V ( π ) − ( P$_{n}$ − P ) f$_{π}$ ∣ . (5) 5 2.1 Preliminary 2 Main Results f$_{π}$ ( o ) ≡ 1 { a = π ( x ) } P ( A = a | X = x ) ( y − E [ Y | A = a, X = x ]) + E [ Y | A = π ( x ) , X = x ] − V ( π ) . (3) F ≡ { f$_{π}$ : π ∈ Π } ⊂ L $^{2}$( P ) (4)