Multi-Objective Linear Reinforcement Learning with Lexicographic Rewards

Abstract

Reinforcement Learning (RL) with linear transition kernels and reward functions has recently attracted growing attention due to its computational efficiency and theoretical advancements. However, prior theoretical research in RL has primarily focused on single-objective problems, resulting in limited theoretical development for multi-objective reinforcement learning (MORL). To bridge this gap, we examine MORL under lexicographic reward structures, where rewards comprise $m$ hierarchically ordered objectives. In this framework, the agent the agent maximizes objectives sequentially, prioritizing the highest-priority objective before considering subsequent ones. We introduce the first MORL algorithm with provable regret guarantees. For any objective $i \in \\{1, 2, \ldots, m\\}$, our algorithm achieves a regret bound of $\widetilde{O}(\Lambda^i(\lambda) \cdot \sqrt{d^2H^4 K})$, where $\Lambda^i(\lambda) = 1 + \lambda + \cdots + \lambda^{i-1}$, $\lambda$ quantifies the trade-off between conflicting objectives, $d$ is the feature dimension, $H$ is the episode length, and $K$ is the number of episodes. Furthermore, our algorithm can be applied in the misspecified setting, where the regret bound for the $i$-th objective becomes $\widetilde{O}(\Lambda^i(\lambda)\cdot(\sqrt{d^2H^4K}+\epsilon dH^2K))$, with $\epsilon$ denoting the degree of misspecification.