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

원문 초록

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 $H=(1-\gamma)^{-1}$, we establish the constant-confidence minimax sample complexity $\widetilde\Theta\!\left( \frac{SAH^3}{\varepsilon^2}\min\{S,\log(em)\} \right)$, 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 optimal-value increments over covers of the reward hull. A one-sided Bellman variance bound controls signed value differences under a second transition kernel, preserving the $H^3\varepsilon^{-2}$ dependence. A complementary uniform guarantee for all bounded rewards yields the state-dimension saturation. We establish lower bounds matching up to logarithmic factors through a rare-exit construction with subset rewards and action encoding. These bounds apply to adaptive simulator queries and arbitrary randomized history-dependent output policies, while the upper bounds are attained by deterministic stationary policies.