Regret Bounds for Reinforcement Learning from Multi-Source Imperfect Preferences

Reinforcement learning from human feedback (RLHF) replaces hard-to-specify rewards with pairwise trajectory preferences, yet regret-oriented theory often assumes that preference labels are generated consistently from a single ground-truth objective. In practical RLHF systems, however, feedback is typically \emph{multi-source} (annotators, experts, reward models, heuristics) and can exhibit systematic, persistent mismatches due to subjectivity, expertise variation, and annotation/modeling artifacts. We study episodic RL from \emph{multi-source imperfect preferences} through a cumulative imperfection budget: for each source, the total deviation of its preference probabilities from an ideal oracle is at most $\omega$ over $K$ episodes. We propose a unified algorithm with regret $\tilde{O}(\sqrt{K/M}+\omega)$, which exhibits a best-of-both-regimes behavior: it achieves $M$-dependent statistical gains when imperfection is small (where $M$ is the number of sources), while remaining robust with unavoidable additive dependence on $\omega$ when imperfection is large. We complement this with a lower bound $\tilde{\Omega}(\max\{\sqrt{K/M},\omega\})$, which captures the best possible improvement with respect to $M$ and the unavoidable dependence on $\omega$, and a counterexample showing that na\"ively treating imperfect feedback as oracle-consistent can incur regret as large as $\tilde{\Omega}(\min\{\omega\sqrt{K},K\})$. Technically, our approach involves imperfection-adaptive weighted comparison learning, value-targeted transition estimation to control hidden feedback-induced distribution shift, and sub-importance sampling to keep the weighted objectives analyzable, yielding regret guarantees that quantify when multi-source feedback provably improves RLHF and how cumulative imperfection fundamentally limits it.