  In this paper, we demonstrate how to learn the objective function of a
decision-maker while only observing the problem input data and the
decision-maker's corresponding decisions over multiple rounds. We present exact
algorithms for this online version of inverse optimization which converge at a
rate of $ \mathcal{O}(1/\sqrt{T}) $ in the number of observations~$T$ and
compare their further properties. Especially, they all allow taking decisions
which are essentially as good as those of the observed decision-maker already
after relatively few iterations, but are suited best for different settings
each. Our approach is based on online learning and works for linear objectives
over arbitrary feasible sets for which we have a linear optimization oracle. As
such, it generalizes previous approaches based on KKT-system decomposition and
dualization. We also introduce several generalizations, such as the approximate
learning of non-linear objective functions, dynamically changing as well as
parameterized objectives and the case of suboptimal observed decisions. When
applied to the stochastic offline case, our algorithms are able to give
guarantees on the quality of the learned objectives in expectation. Finally, we
show the effectiveness and possible applications of our methods in indicative
computational experiments.
