Adaptive Exploration for Multi-Reward Multi-Policy Evaluation

We study the policy evaluation problem in an online multi-reward multi-policy discounted setting, where multiple reward functions must be evaluated simultaneously for different policies. We adopt an $(\epsilon,\delta)$-PAC perspective to achieve $\epsilon$-accurate estimates with high confidence across finite or convex sets of rewards, a setting that has not been investigated in the literature. Building on prior work on Multi-Reward Best Policy Identification, we adapt the MR-NaS exploration scheme to jointly minimize sample complexity for evaluating different policies across different reward sets. Our approach leverages an instance-specific lower bound revealing how the sample complexity scales with a measure of value deviation, guiding the design of an efficient exploration policy. Although computing this bound entails a hard non-convex optimization, we propose an efficient convex approximation that holds for both finite and convex reward sets. Experiments in tabular domains demonstrate the effectiveness of this adaptive exploration scheme.