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 non-linear 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 non-linear 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.

翻译：我们研究公平多目标强化学习，其中智能体必须学习一种策略，以在向量值奖励的多个维度上同时获得高回报。受公平资源分配文献的启发，我们将此建模为期望福利最大化问题，其中采用某种关于长期累积奖励向量的非线性公平福利函数。此类函数的一个典型例子是纳什社会福利（即几何平均值），其对数变换也被称为比例公平目标。我们证明，即使是在表格情形下，优化期望纳什社会福利的近似最优解在计算上也是不可行的。尽管如此，我们提出了一种新颖的Q-learning改进方法，该方法结合了非线性标量化的学习更新和非平稳动作选择，以学习优化非线性福利函数的有效策略。我们证明该算法具有收敛性，并通过实验表明，在处理纳什社会福利目标时，我们的方法优于基于线性标量化、最优线性标量化混合或平稳动作选择的技术。