Multi-Objective Reinforcement Learning: Convexity, Stationarity and Pareto Optimality

In recent years, single-objective reinforcement learning (SORL) algorithms have received a significant amount of attention and seen some strong results. However, it is generally recognized that many practical problems have intrinsic multi-objective properties that cannot be easily handled by SORL algorithms. Although there have been many multi-objective reinforcement learning (MORL) algorithms proposed, there has been little recent exploration of the fundamental properties of the spaces we are learning in. In this paper, we perform a rigorous analysis of policy induced value functions and use the insights to distinguish three views of Pareto optimality. The results imply the convexity of the induced value function's range for stationary policies and suggest that any point of its Pareto front can be achieved by training a policy using linear scalarization (LS). We show the problem that leads to the suboptimal performance of LS can be solved by adding strongly concave terms to the immediate rewards, which motivates us to propose a new vector reward-based Q-learning algorithm, CAPQL. Combined with an actor-critic formulation, our algorithm achieves state-of-the-art performance on multiple MuJoCo tasks in the preference agnostic setting. Furthermore, we empirically show that, in contrast to other LS-based algorithms, our approach is significantly more stable, achieving similar results across various random seeds.