  An agnostic PAC learning algorithm finds a predictor that is competitive with
the best predictor in a benchmark hypothesis class, where competitiveness is
measured with respect to a given loss function. However, its predictions might
be quite sub-optimal for structured subgroups of individuals, such as protected
demographic groups. Motivated by such fairness concerns, we study "multi-group
agnostic PAC learnability": fixing a measure of loss, a benchmark class $\H$
and a (potentially) rich collection of subgroups $\G$, the objective is to
learn a single predictor such that the loss experienced by every group $g \in
\G$ is not much larger than the best possible loss for this group within $\H$.
Under natural conditions, we provide a characterization of the loss functions
for which such a predictor is guaranteed to exist. For any such loss function
we construct a learning algorithm whose sample complexity is logarithmic in the
size of the collection $\G$. Our results unify and extend previous positive and
negative results from the multi-group fairness literature, which applied for
specific loss functions.
