  Inverse multiobjective optimization provides a general framework for the
unsupervised learning task of inferring parameters of a multiobjective decision
making problem (DMP), based on a set of observed decisions from the human
expert. However, the performance of this framework relies critically on the
availability of an accurate DMP, sufficient decisions of high quality, and a
parameter space that contains enough information about the DMP. To hedge
against the uncertainties in the hypothetical DMP, the data, and the parameter
space, we investigate in this paper the distributionally robust approach for
inverse multiobjective optimization. Specifically, we leverage the Wasserstein
metric to construct a ball centered at the empirical distribution of these
decisions. We then formulate a Wasserstein distributionally robust inverse
multiobjective optimization problem (WRO-IMOP) that minimizes a worst-case
expected loss function, where the worst case is taken over all distributions in
the Wasserstein ball. We show that the excess risk of the WRO-IMOP estimator
has a sub-linear convergence rate. Furthermore, we propose the semi-infinite
reformulations of the WRO-IMOP and develop a cutting-plane algorithm that
converges to an approximate solution in finite iterations. Finally, we
demonstrate the effectiveness of our method on both a synthetic multiobjective
quadratic program and a real world portfolio optimization problem.
