If empirical process conditions are imposed on the estimators for the reward regression, E [ Y | A, X ], and the action mechanism, P ( A | X ), then ̂ V could be defined using estimat- ing equations [15] or targeted minimum loss-based estimation (TMLE) [16, 17]. Without imposing empirical process conditions, one could use cross-validated estimating equations [18, 19], now also called double machine learning [20], or a cross-validated TMLE [21]. Note that cross-validated estimating equations in this scenario represent a special case of a cross- validated TMLE, where one uses the squared cross-validated efficient influence function as loss [22]. For any of these listed approaches, obtaining the above uniformity over π is straightforward if Π is a Donsker class. Our first objective will be to establish a faster than n − 1 / 2 rate of regret decay for a value- based estimator ̂ π of an optimal policy, given by any ERM ̂ π ∈ Π satisfying ̂ V $^{(}$̂ π ) ≥ sup π ∈ Π ̂ V ( π ) − o$_{P}$ (Rem$_{n}$) . (6) Note that (6) is a requirement on the behavior of the optimization algorithm on the realized sample, rather than a statistical or probabilistic condition. The o$_{P}$ (Rem$_{n}$) term allows us to obtain an approximate solution to the ERM problem, which is useful because the optimization on the right is non-concave. Π ⋆ ≡ { π ⋆ ∈ Π : V ( π $^{⋆}$) = V $^{⋆}$} (7) = { π ⋆ ∈ Π : R ( π $^{⋆}$) = 0 } ⊂ { π ⋆ ∈ Π : R ( π $^{⋆}$) ≤ 0 } 6 2.2 Empirical Risk Minimizers We now present a result that is similar to the groundbreaking results for ERMs based on general losses given in [11]. In Section 3.1, we discuss how our results relate to those in [11]. The upcoming theorem uses Π to denote the closure of Π under L $^{2}$( P ) norm. The set