estimation strategy can sometimes yield better estimates than a value-based strategy [5]. Let γ ( X ) ≡ E [ Y | A = 1 , X ] − E [ Y | A = − 1 , X ] denote the conditional average action effect. In this subsection, we assume that Π is an unrestricted class, i.e. , that Π contains all functions mapping X to {− 1 , 1 } . Audibert and Tsybakov [10] presented the surprising fact that plug-in classifiers can attain much faster rates (faster than n − $^{1}$). In our setting, the plug-in policy that we will study first defines an estimator ̂ γ of γ , and then determines the action to undertake based on the sign of the resulting estimate. Formally, ̂ π is given by Extensions of the result of [10] have previously been presented in both the reinforcement learning literature [6] and in the optimal individualized treatment literature [7, 8]. We therefore omit any proof, and refer the reader to [10, Lemma 5.2] for further details. The upcoming theorem uses ‖·‖ to denote the L $^{2}$( P ) norm and ‖·‖$_{∞}$ to denote the L $^{∞}$( P ) norm. Let ≲ denote “less than or equal to up to a universal positive multiplicative constant” and consider the following “margin assumption”: MA) For some α > 0, P (0 < | γ ( X ) | ≤ t ) ≲ t α for all t > 0. If $^{‖}$̂ γ − γ ‖$_{∞}$ = o$_{P}$ (1) , then 0 ≤ R $^{(}$̂ π ) ≲ $^{‖}$̂ γ − γ ‖ ∞ . We now briefly describe this result, though we note that it has been discussed thoroughly elsewhere [10, 7, 8]. Note that MA does not place any restriction on the decision boundary { x ∈ X : γ ( x ) = 0 } where no action is superior to the other, but rather only places a restriction on the probability that γ ( X ) is near zero. Theorem 2. Suppose MA holds. If $^{‖}$̂ γ − γ ‖ = o$_{P}$ (1) , then 0 ≤ R $^{(}$̂ π ) ≲ $^{‖}$̂ γ − γ ‖ 2(1+ α ) / (2+ α $^{)}$. If γ − γ ‖$_{∞}$ = o$_{P}$ (1) , then 0 ≤ R π ) ≲ γ − γ ‖ 1+ α ∞ . If α = 1 and γ ( X ) is absolutely continuous (under P ), then MA corresponds to γ ( X ) having bounded density near zero. The case that α = 1 is of particular interest for the sup-norm result because then the regret bound is quadratic in the rate of convergence of ̂ γ to γ . As α → 0, more mass is placed near the decision boundary (zero) and the above result 8 ̂ π ( x ) ≡ 1 $^{{}$̂ γ ( x ) > 0 } − 1 $^{{}$̂ γ ( x ) ≤ 0 } .