  We address an inherent difficulty in welfare-theoretic fair machine learning
by proposing an equivalently axiomatically-justified alternative and studying
the resulting computational and statistical learning questions. Welfare metrics
quantify overall wellbeing across a population of one or more groups, and
welfare-based objectives and constraints have recently been proposed to
incentivize fair machine learning methods to produce satisfactory solutions
that consider the diverse needs of multiple groups. Unfortunately, many
machine-learning problems are more naturally cast as loss minimization tasks,
rather than utility maximization, which complicates direct application of
welfare-centric methods to fair machine learning. In this work, we define a
complementary measure, termed malfare, measuring overall societal harm (rather
than wellbeing), with axiomatic justification via the standard axioms of
cardinal welfare. We then cast fair machine learning as malfare minimization
over the risk values (expected losses) of each group. Surprisingly, the axioms
of cardinal welfare (malfare) dictate that this is not equivalent to simply
defining utility as negative loss. Building upon these concepts, we define
fair-PAC (FPAC) learning, where an FPAC learner is an algorithm that learns an
$\varepsilon$-$\delta$ malfare-optimal model with bounded sample complexity,
for any data distribution, and for any (axiomatically justified) malfare
concept. Finally, we show broad conditions under which, with appropriate
modifications, standard PAC-learners may be converted to FPAC learners. This
places FPAC learning on firm theoretical ground, as it yields statistical and
computational efficiency guarantees for many well-studied machine-learning
models, and is also practically relevant, as it democratizes fair ML by
providing concrete training algorithms and rigorous generalization guarantees
for these models
