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O , while still denoting ( O, W ) ∼ P $^{g}$. A balanced design corresponds to P $^{b}$, where b : {− 1 , 1 } × X → [0 , 1] only takes the value 1/2. ✷ 

Optimal policy estimation. We now define set of candidate policies Π. In particular Π is the set of policies such that, for each π ∈ Π, there exists β ∈ ∪$_{n}$$_{≥}$$_{1}$ B$_{n}$ for which 

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where the choice to let A = 1 when Q$_{β}$ (1 , X ) = Q$_{β}$ ( − 1 , X ) in the second equality is a convention. The value V ( π ) of π ∈ Π is still given by (1) and its (within class Π) regret R ( π ) by (2). 

Recall the definition of F characterized by (3) and (4). Let ˜ G n ∈ ℓ $^{∞}$( F ) be such that, for each f ∈ F , 

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where we use the fact that g$_{i}$ is deterministic when one conditions on O$_{1}$, . . . , O$_{i}$$_{−}$$_{1}$ ( i.e. , on nothing when i = 1). By exploiting the martingale structure of each n 1 / 2 ˜ G $_{n}$f , it is possible to show that ˜ G n satisfies (12) under mild conditions (expressed in terms of the uniform entropy integral) on {Q$_{n}$}$_{n}$$_{≥}$$_{1}$ [28, Lemma B.3]. Theorem 4 yields a corollary for our sequential design setting. We discuss the conditions of the corollary following its statement. 

Corollary 14. In the context of Section 4.4, suppose that {Q$_{n}$}$_{n}$$_{≥}$$_{1}$ is chosen in such a way that Π is separable and J$_{1}$ ( η$_{n}$, Π) = o (1) for any η$_{n}$ = o (1) , where J$_{1}$ ( η, Π) is the uniform entropy integral evaluated at η of Π wrt the envelope function constantly equal to one. Moreover, suppose that ̂ π satisfies (13) with r$_{n}$ ≡ n 1 / 2 and that 1 / 2 

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