  We consider the setting of stochastic bandit problems with a continuum of
arms. We first point out that the strategies considered so far in the
literature only provided theoretical guarantees of the form: given some tuning
parameters, the regret is small with respect to a class of environments that
depends on these parameters. This is however not the right perspective, as it
is the strategy that should adapt to the specific bandit environment at hand,
and not the other way round. Put differently, an adaptation issue is raised. We
solve it for the special case of environments whose mean-payoff functions are
globally Lipschitz. More precisely, we show that the minimax optimal orders of
magnitude $L^{d/(d+2)} \, T^{(d+1)/(d+2)}$ of the regret bound against an
environment $f$ with Lipschitz constant $L$ over $T$ time instances can be
achieved without knowing $L$ or $T$ in advance. This is in contrast to all
previously known strategies, which require to some extent the knowledge of $L$
to achieve this performance guarantee.
