  We study fair multi-objective reinforcement learning in which an agent must
learn a policy that simultaneously achieves high reward on multiple dimensions
of a vector-valued reward. Motivated by the fair resource allocation
literature, we model this as an expected welfare maximization problem, for some
nonlinear fair welfare function of the vector of long-term cumulative rewards.
One canonical example of such a function is the Nash Social Welfare, or
geometric mean, the log transform of which is also known as the Proportional
Fairness objective. We show that even approximately optimal optimization of the
expected Nash Social Welfare is computationally intractable even in the tabular
case. Nevertheless, we provide a novel adaptation of Q-learning that combines
nonlinear scalarized learning updates and non-stationary action selection to
learn effective policies for optimizing nonlinear welfare functions. We show
that our algorithm is provably convergent, and we demonstrate experimentally
that our approach outperforms techniques based on linear scalarization,
mixtures of optimal linear scalarizations, or stationary action selection for
the Nash Social Welfare Objective.
