≤ 1 2 E [ | π$_{1}$ ( X ) − π$_{2}$ ( X ) | P ( A | X ) 1 { m < Y ≤ m + ϵ } ] ≲ E [ | π$_{1}$ ( X ) − π$_{2}$ ( X ) | 1 { m < Y ≤ m + ϵ } ] = E [ | π$_{1}$ ( X ) − π$_{2}$ ( X ) | P ( m < Y ≤ m + ϵ | X )] ≤ Lϵ E [ | π$_{1}$ ( X ) − π$_{2}$ ( X ) | ] ≲ ϵ ‖ π$_{1}$ − π$_{2}$ ‖ . Returning to the first display completes the proof. Corollary B.9. Under the conditions of Lemmas 9 and B.8, and the additional assumption (16), the map π ↦→ ˙ F$_{π}$ ( V ( π )) is uniformly continuous on Π with respect to ‖·‖ . Proof of Corollary B.9. Set π$_{1}$, π$_{2}$ ∈ Π. By Lemma 9, we can choose π$_{2}$ sufficiently close to π$_{1}$ in L $^{2}$( P ) (where “sufficiently close” does not depend on the choice of π$_{1}$ ) so that |V ( π$_{1}$ ) − V ( π$_{2}$ ) | ≤ c , where c is the constant from (15). For all such choices of π$_{1}$, π$_{2}$ , where the constant on the right does not depend on π$_{1}$, π$_{2}$ . The final bound used Lemma B.8 and (16). As ω (0) = 0 and ω is continuous at zero, one can choose ‖ π$_{1}$ − π$_{2}$ ‖ sufficiently small so that the right-hand side is less than any ϵ > 0. As “sufficiently small” does not depend on the choice of π$_{1}$ , π ↦→ ˙ F$_{π}$ ( V ( π )) is uniformly continuous on Π with respect to ‖·‖ . References 35 [1] S Athey and S Wager. Efficient Policy Learning. arXiv preprint arXiv:1702.02896 , 2017. [2] Y Zhao, D Zeng, A Rush, and M Kosorok. Estimating individual treatment rules using outcome weighted learning. J. Am. Stat. Assoc. , 107:1106–1118, 2012. [3] B Zhang, A A Tsiatis, M Davidian, M Zhang, and E Laber. Estimating optimal treat- ment regimes from a classification perspective. Stat , 68(1):103–114, 2012. ∣ ˙ F$_{π}$$_{2}$ ( V ( π$_{2}$ )) − ˙ F$_{π}$$_{1}$ ( V ( π$_{1}$ )) ∣ ≤ ∣ ˙ F$_{π}$$_{2}$ ( V ( π$_{2}$ )) − ˙ F$_{π}$$_{1}$ ( V ( π$_{2}$ )) ∣ + ∣ ˙ F$_{π}$$_{1}$ ( V ( π$_{2}$ )) − ˙ F$_{π}$$_{1}$ ( V ( π$_{1}$ )) ∣ ≲ ‖ π$_{1}$ − π$_{2}$ ‖ + ω ( ‖ π$_{1}$ − π$_{2}$ ‖ ) ,