term, one could use simlar arguments to those used in [11, Section 4] to control the regret (though, the proofs in our work are self-contained). We were not able to find a similar result in the classification literature that shows that Π Donsker suffices to attain a fast rate on misclassification error (the classification analogue of our regret). A result that makes a similarly weak complexity requirement (only uses a Donsker condition) was given for general result for ERMs in [26, Theorem 4.5]. In particular, that result shows that, if Π is Donsker and Π ⋆ is a singleton, then one attains a fast rate. In the policy learning context, this is a weaker result than what we have shown: Π Donsker implies a fast rate, without any assumption on the cardinality of Π $^{⋆}$. We note also that our result could readily be extended to standard classification problems: the lack of a remainder term Rem$_{n}$ only makes the problem easier. Tightening Theorem 1 would require careful consideration of the rate of convergence of two o$_{P}$ ( n − 1 / $^{2}$) terms, namely Rem$_{n}$ and EP$_{n}$. On the one hand, if one uses a cross-validated estimator, then the rate of Rem$_{n}$ is typically dominated by the rate of a doubly robust term, which is in turn upper bounded by the product of the L $^{2}$( P ) norm rates of convergence of the estimated action mechanism and reward regression. It is thus sufficient to assume that this product is o$_{P}$ ( n − 1 / $^{2}$) to guarantee that Rem$_{n}$ = o$_{P}$ ( n − 1 / $^{2}$). It is worth noting that the product is O$_{P}$ ( n − $^{1}$) in a well-specified parametric model. On the other hand, the magnitude of EP$_{n}$ is controlled by both the size of class Π and the behavior of γ near the decision boundary, hence the need for a margin assumption like MA. We do not believe there is a general ordering between the rate of convergence of Rem$_{n}$ and EP$_{n}$ that applies across all problems, and so it does not appear that the size of Π nor the use of an efficient choice of influence function fully determines the rate of regret decay of ̂ π , even when Rem$_{n}$ = o$_{P}$ ( n − 1 / $^{2}$). A notable exception to this lack of strict ordering between Rem$_{n}$ and EP$_{n}$ occurs when the action mechanism is known: in this case, the size of Π and 3.2 Tightening Theorem 1 10