  The most prominent feedback models for the best expert problem are the full
information and bandit models. In this work we consider a simple feedback model
that generalizes both, where on every round, in addition to a bandit feedback,
the adversary provides a lower bound on the loss of each expert. Such lower
bounds may be obtained in various scenarios, for instance, in stock trading or
in assessing errors of certain measurement devices. For this model we prove
optimal regret bounds (up to logarithmic factors) for modified versions of
Exp3, generalizing algorithms and bounds both for the bandit and the
full-information settings. Our second-order unified regret analysis simulates a
two-step loss update and highlights three Hessian or Hessian-like expressions,
which map to the full-information regret, bandit regret, and a hybrid of both.
Our results intersect with those for bandits with graph-structured feedback, in
that both settings can accommodate feedback from an arbitrary subset of experts
on each round. However, our model also accommodates partial feedback at the
single-expert level, by allowing non-trivial lower bounds on each loss.
