  We study best-arm identification with fixed confidence in bandit models with
graph smoothness constraint. We provide and analyze an efficient gradient
ascent algorithm to compute the sample complexity of this problem as a solution
of a non-smooth max-min problem (providing in passing a simplified analysis for
the unconstrained case). Building on this algorithm, we propose an
asymptotically optimal strategy. We furthermore illustrate by numerical
experiments both the strategy's efficiency and the impact of the smoothness
constraint on the sample complexity. Best Arm Identification (BAI) is an
important challenge in many applications ranging from parameter tuning to
clinical trials. It is now very well understood in vanilla bandit models, but
real-world problems typically involve some dependency between arms that
requires more involved models. Assuming a graph structure on the arms is an
elegant practical way to encompass this phenomenon, but this had been done so
far only for regret minimization. Addressing BAI with graph constraints
involves delicate optimization problems for which the present paper offers a
solution.
