content_image/314062.jpg content_image/314072.jpg
Returning to the first display completes the proof. 

<img src='content_image/314062.jpg'>

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$_{2}$ , 

<img src='content_image/314072.jpg'>

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 

[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. 

