Language model (LM) alignment improves model outputs to reflect human preferences while preserving the capabilities of the base model. The most common alignment approaches are (i) reinforcement learning, which maximizes the expected reward under a KL-divergence constraint, and (ii) best-of-NNN alignment, which selects the highest-reward output among NNN independent samples. Despite their widespread use, the fundamental limits of reward improvement under a KL budget remain poorly understood. We characterize the information-theoretic limits of KL-regularized alignment by deriving the maximum…