Breaking the Bias Barrier in Concave Multi-Objective Reinforcement Learning

TL;DR

This paper addresses the bias in policy gradient methods for concave multi-objective reinforcement learning, proposing an algorithm with optimal sample complexity guarantees by controlling gradient bias through advanced estimators.

Contribution

It introduces a Natural Policy Gradient algorithm with a multi-level Monte Carlo estimator to achieve optimal sample complexity in concave multi-objective RL.

Findings

Achieves $ ilde{O}(rac{1}{ ext{epsilon}^2})$ sample complexity for $ ext{epsilon}$-optimal policies.

Identifies the bias barrier in existing methods and overcomes it with MLMC estimator.

Shows second-order smoothness cancels bias, enabling simpler algorithms to reach optimal rates.

Abstract

While standard reinforcement learning optimizes a single reward signal, many applications require optimizing a nonlinear utility $f(J_1^\pi,\dots,J_M^\pi)$ over multiple objectives, where each $J_m^\pi$ 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 $\partial f(J^\pi)$, 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 $\widetilde{\mathcal{O}}(\epsilon^{-4})$ 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 $\widetilde{\mathcal{O}}(\epsilon^{-2})$ sample complexity for computing an $\epsilon$-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 $\widetilde{\mathcal{O}}(\epsilon^{-2})$ rate without MLMC. Our results provide the first optimal sample complexity guarantees for concave multi-objective reinforcement learning under policy-gradient methods.