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...s for the regressogram defined above. Upper confidence bounds (UCB) policies are known to perform optimally in the traditional two armed bandit problem, i.e., without covariates (Lai & Robbins, 1985; =-=Auer et al., 2002-=-). The index of each arm is computed as the sum of the average past reward and a stochastic term accounting for the deviations of the observed average reward from the true average reward. In the UCBog...

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...pper confidence bounds for the regressogram defined above. Upper confidence bounds (UCB) policies are known to perform optimally in the traditional two armed bandit problem, i.e., without covariates (=-=Lai & Robbins, 1985-=-; Auer et al., 2002). The index of each arm is computed as the sum of the average past reward and a stochastic term accounting for the deviations of the observed average reward from the true average r...

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...plicit. In view of Lemma 3.1, it is sufficient to prove (14). To do so we reduce our problem to a hypothesis testing problems; an approach this is quite standard in the nonparametric literature, cf. (=-=Tsybakov, 2009-=-, the joint distribution of the collection of Chapter 2). For any policy π, and any t = 1, . . . , n, denote by IP t π,f pairs (X1, Y (π1(X1)) 1 ), . . . , (Xt, Y (πt(Xt)) t ) where IE[Y (1)|X] = f(X)...

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...d margin condition, as it has been come be known in the full information setup; cf. Tsybakov (2004). In that setting, it has been shown to critically affect the complexity of classification problems (=-=Tsybakov, 2004-=-; Boucheron et al., 2005; Audibert & Tsybakov, 2007). In the bandit setup, this condition encodes the “separation” between the functions that describe the arms’ responses and was originally studied by...

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... number of bins M as a function of the horizon n, while in practice one does not have foreknowledge of this value. This limitation can be easily circumvented by using the so-called doubling argument (=-=Cesa-Bianchi & Lugosi, 2006-=-) which consists of “reseting” the game at times 2 k , k = 1, 2, . . . The reader will note that when α = 1 there is a potentially superfluous log n factor appearing in the upper bound in the theorem....

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... quantity. The second condition imposed gives a lower bound on this function though in a weaker globalsense. It is closely related to the margin condition employed in classification (Tsybakov, 2004; =-=Mammen & Tsybakov, 1999-=-), which drives the terminology employed here. MARGIN CONDITION. We say that the machine satisfies the margin condition with parameter α if there exists δ0 ∈ (0, 1), Cδ > 0 such that [ ] (1) (2) 0 < |...

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...on, as it has been come be known in the full information setup; cf. Tsybakov (2004). In that setting, it has been shown to critically affect the complexity of classification problems (Tsybakov, 2004; =-=Boucheron et al., 2005-=-; Audibert & Tsybakov, 2007). In the bandit setup, this condition encodes the “separation” between the functions that describe the arms’ responses and was originally studied by Goldenshluger and Zeevi...

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...be known in the full information setup; cf. Tsybakov (2004). In that setting, it has been shown to critically affect the complexity of classification problems (Tsybakov, 2004; Boucheron et al., 2005; =-=Audibert & Tsybakov, 2007-=-). In the bandit setup, this condition encodes the “separation” between the functions that describe the arms’ responses and was originally studied by Goldenshluger and Zeevi (2009) in the one armed ba...