  We consider the bandit-based framework for diversity-preserving
recommendations introduced by Celis et al. (2019), who approached it in the
case of a polytope mainly by a reduction to the setting of linear bandits. We
design a UCB algorithm using the specific structure of the setting and show
that it enjoys a bounded distribution-dependent regret in the natural cases
when the optimal mixed actions put some probability mass on all actions (i.e.,
when diversity is desirable). The regret lower bounds provided show that
otherwise, at least when the model is mean-unbounded, a $\ln T$ regret is
suffered. We also discuss an example beyond the special case of polytopes.
