  Modern machine learning tasks often require considering not just one but
multiple objectives. For example, besides the prediction quality, this could be
the efficiency, robustness or fairness of the learned models, or any of their
combinations. Multi-objective learning offers a natural framework for handling
such problems without having to commit to early trade-offs. Surprisingly,
statistical learning theory so far offers almost no insight into the
generalization properties of multi-objective learning. In this work, we make
first steps to fill this gap: we establish foundational generalization bounds
for the multi-objective setting as well as generalization and excess bounds for
learning with scalarizations. We also provide the first theoretical analysis of
the relation between the Pareto-optimal sets of the true objectives and the
Pareto-optimal sets of their empirical approximations from training data. In
particular, we show a surprising asymmetry: all Pareto-optimal solutions can be
approximated by empirically Pareto-optimal ones, but not vice versa.
