Improved Bounds for Private and Robust Alignment

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-$N$ alignment, which selects the highest-reward output among $N$ 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 achievable expected reward gain for a fixed KL-divergence budget. Our first result provides a closed-form expression for the optimal reward improvement, governed by a Jeffreys divergence term rather than the $\sqrt{\texttt{KL}}$ used in prior analyses. We further reformulate this expression as a covariance under the base model, yielding a practical estimator that predicts achievable alignment gains from base model samples alone. We extend our analysis to the proxy reward setting, showing that the gap between ideal and proxy alignment (reward hacking) grows with the magnitude of reward error and when the KL penalty factor decreases. We then prove that reward ensembling mitigates reward hacking, providing a theoretical justification for this technique used in practice. Empirically, we compute the KL-reward Pareto frontier for two tasks for LMs, safety and summarization, and show that best-of-$N$ closely approaches the theoretical limit, while PPO and GRPO remain substantially suboptimal. Our theoretical results shed light on several empirically observed phenomena in the alignment literature and suggest that algorithmic improvements are needed to achieve optimal alignment without high inference costs.

Existing robust preference optimization for language-model alignment mainly studies pairwise supervision and places robustness at the dataset, prompt, or preference-pair level. We instead study listwise preference optimization under ranking-label uncertainty: given a prompt and a candidate list, the observed ranking over that list may be ambiguous due to annotator inconsistency, near-ties, lossy rankwise feedback, or reward-model noise. We propose a pointwise total-variation robust Plackett--Luce objective that directly robustifies the ranking label conditional on the candidate list. The robust loss admits an exact decomposition into the nominal PL loss plus a worst-case PL correction, and the worst-case ranking is obtained by sorting current implicit scores in ascending order, reducing the inner maximization from $K!$ enumeration to $O(K\log K)$. This tractable structure yields strong offline and online optimization guarantees. In the offline fixed-list setting, the robust objective is convex and projected stochastic subgradient reaches global $ε$-suboptimality with $O(ε^{-2})$ sample complexity. In the online policy-induced setting, where candidate lists are generated by the current policy, we establish weak convexity and $\widetilde O(ε^{-2})$ Moreau-envelope stationarity. Experiments in offline LLM alignment show that the proposed robust correction largely preserves performance under clean labels and improves robustness under noise. In online alignment, it makes reward-model-ranked candidate expansion more reliable and improves both reward-model and external GPT-4 judge metrics.

Personalized alignment aims to adapt large language models to heterogeneous user preferences, yet the precise theoretical conditions for its statistical efficiency have not been formally established. This paper characterizes the conditions under which personalized alignment achieves O(1) online regret and log(1/epsilon) offline sample complexity. We show that these optimal rates depend on a specific user-diversity condition: the population of user-specific heads must span the latent reward directions that can alter the optimal response. We prove that this condition is both necessary and sufficient. When it holds, simple greedy algorithms achieve benchmark efficiency; when it fails, every learner in a natural admissible class incurs at least logarithmic regret. Our results identify user diversity as the fundamental driver of personalized identifiability.