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In this paper, we study sequential testing problems with \emph{overlapping}hypotheses. We first focus on the simple problem of assessing if the mean $\mu$of a Gaussian distribution is smaller or larger than a fixed $\epsilon>0$; if$\mu\in(-\epsilon,\epsilon)$, both answers are considered to be correct. Then,we consider PAC-best arm identification in a bandit model: given $K$probability distributions on $\mathbb{R}$ with means $\mu_1,\dots,\mu_K$, wederive the asymptotic complexity of identifying, with risk at most $\delta$, anindex $I\in\{1,\dots,K\}$ such that $\mu_I\geq \max_i\mu_i -\epsilon$. Weprovide non-asymptotic bounds on the error of a parallel General LikelihoodRatio Test, which can also be used for more general testing problems. Wefurther propose lower bound on the number of observation needed to identify acorrect hypothesis. Those lower bounds rely on information-theoretic arguments,and specifically on two versions of a change of measure lemma (a high-levelform, and a low-level form) whose relative merits are discussed.

Non-Asymptotic Sequential Tests for Overlapping Hypotheses and application to near optimal arm identification in bandit models