  We study the problem of the identification of m arms with largest means under
a fixed error rate $\delta$ (fixed-confidence Top-m identification), for
misspecified linear bandit models. This problem is motivated by practical
applications, especially in medicine and recommendation systems, where linear
models are popular due to their simplicity and the existence of efficient
algorithms, but in which data inevitably deviates from linearity. In this work,
we first derive a tractable lower bound on the sample complexity of any
$\delta$-correct algorithm for the general Top-m identification problem. We
show that knowing the scale of the deviation from linearity is necessary to
exploit the structure of the problem. We then describe the first algorithm for
this setting, which is both practical and adapts to the amount of
misspecification. We derive an upper bound to its sample complexity which
confirms this adaptivity and that matches the lower bound when $\delta$
$\rightarrow$ 0. Finally, we evaluate our algorithm on both synthetic and
real-world data, showing competitive performance with respect to existing
baselines.
