Breaking the Bias Barrier in Concave Multi-Objective Reinforcement Learning

While standard reinforcement learning optimizes a single reward signal, many applications require optimizing a nonlinear utility $f(J_1^π,dots,J_M^π)$ over multiple objectives, where each $J_m^π$ denotes the expected discounted return of a distinct reward function. A common approach is concave scalarization, which captures important trade-offs such as fairness and risk sensitivity. However, nonlinear scalarization introduces a fundamental challenge for policy gradient methods: the gradient depends on $partial f(J^π)$, while in practice only empirical return estimates $hat J$ are available. Because $f$ is nonlinear, the plug-in estimator is biased ($mathbb{E}[partial f(hat J)] neq partial f(mathbb{E}[hat J])$), leading to persistent gradient bias that degrades sample complexity.
In this work we identify and overcome this bias barrier in concave-scalarized multi-objective reinforcement learning. We show that existing policy-gradient methods suffer an intrinsic $widetilde{mathcal{O}}(ε^{-4})$ sample complexity due to this bias. To address this issue, we develop a Natural Policy Gradient (NPG) algorithm equipped with a multi-level Monte Carlo (MLMC) estimator that controls the bias of the scalarization gradient while maintaining low sampling cost. We prove that this approach achieves the optimal $widetilde{mathcal{O}}(ε^{-2})$ sample complexity for computing an $ε$-optimal policy. Furthermore, we show that when the scalarization function is second-order smooth, the first-order bias cancels automatically, allowing vanilla NPG to achieve the same $widetilde{mathcal{O}}(ε^{-2})$ rate without MLMC. Our results provide the first optimal sample complexity guarantees for concave multi-objective reinforcement learning under policy-gradient methods.