Near-Minimax Sample Complexity for Multi-Objective Reinforcement Learning under Generative Models

Yue Wang

In multi-objective reinforcement learning, preferences may be specified only after data collection, requiring one dataset to support a continuum of reward trade-offs. We study finite discounted Markov decision processes with unknown transitions, $S$ states, $A$ actions, and $m$ known reward functions taking values in $[0,1]$, given access to a generative model. The learner must provide an $\varepsilon$-optimal policy for every convex combination of these rewards without further sampling, with a guarantee that holds simultaneously over all preferences and initial states. Writing $H=(1-γ)^{-1}$, we establish the constant-confidence minimax sample complexity $\widetildeΘ\!\left( \frac{SAH^3}{\varepsilon^2}\min\{S,\log(em)\} \right)$, where the suppressed factors are logarithmic in $S,A,H,\varepsilon^{-1}$ and the dependence on $m$ is explicit. Thus, up to these logarithmic factors, the worst-case statistical cost of supporting all preferences grows logarithmically with the number of objectives and saturates at the number of states. The upper bound uses uniform state--action sampling and empirical planning, with reward rounding in the logarithmic-objective regime. Its analysis chains…