  We study the problems of offline and online contextual optimization with
feedback information, where instead of observing the loss, we observe,
after-the-fact, the optimal action an oracle with full knowledge of the
objective function would have taken. We aim to minimize regret, which is
defined as the difference between our losses and the ones incurred by an
all-knowing oracle. In the offline setting, the decision-maker has information
available from past periods and needs to make one decision, while in the online
setting, the decision-maker optimizes decisions dynamically over time based a
new set of feasible actions and contextual functions in each period. For the
offline setting, we characterize the optimal minimax policy, establishing the
performance that can be achieved as a function of the underlying geometry of
the information induced by the data. In the online setting, we leverage this
geometric characterization to optimize the cumulative regret. We develop an
algorithm that yields the first regret bound for this problem that is
logarithmic in the time horizon. Finally, we show via simulation that our
proposed algorithms outperform previous methods from the literature.
