Safety Hacking in Constrained Best-of-$N$ Inference-time Scaling

Inference-time alignment steers a frozen language model during decoding using auxiliary reward signals, avoiding the cost of repeated weight updates. However, existing inference-time alignment methods typically optimize a single scalar score, so explicit safety constraints must either be ignored or encoded through manually tuned penalties. We propose Lagrangian Reward Augmentation (LARA), a general inference-time alignment framework under safety constraints. Starting from a KL-regularized constrained objective with a reward model and a cost model, LARA dualizes the constraint and reduces the optimization problem to a one-dimensional convex problem over a nonnegative dual variable. Estimated on a small calibration set, this dual variable defines an augmented reward that can be used as a drop-in scoring signal within existing inference-time alignment methods. For sequence-level sampling methods, such as Best-of-N reranking, the calibrated dual variable corresponds to the solution of the expected-cost constrained problem. For token-level reward-guided decoding methods, the same construction yields a principled dual-calibrated heuristic rather than an exact constrained-policy guarantee. We evaluate LARA on both sequence-level and token-level inference-time alignment methods, and find that LARA improves the helpfulness-harmlessness tradeoff, with Best-of-N achieving the best performance among inference-time methods, approaching finetuning-based direct alignment baselines.

Best-of-N sampling is a simple inference-time scaling strategy for large language models, yet its behavior under incomplete reward models remains poorly understood. We study Best-of-N as a competition between the extreme reward tails of correct and incorrect responses. This view shows that success depends not only on the probability of sampling a correct response, but also on whether correct responses dominate incorrect ones in the upper tail. Using extreme value theory, we characterize three asymptotic regimes: correct-tail-dominant, incorrect-tail-dominant, and critical explaining both monotonic gains and reward-hacking-induced degradation. We further derive finite-sample scaling laws for representative reward distributions, showing that, under Gaussian rewards, incorrect responses with sufficiently large variance can dominate selection even when their mean reward is lower. Unlike oracle selection, convergence with an incomplete selector can be polynomial rather than exponential. Finally, we propose a lightweight tail-tension criterion to estimate when additional sampling may become harmful. Experiments on synthetic data and real reward model scores are consistent with the theoretically predicted regimes, scaling laws, and diagnostic behavior, suggesting that tail competition provides a unified lens on Best-of-N sampling for verifiable problems.

Inference-time compute scaling has emerged as a powerful paradigm for improving language model performance on a wide range of tasks, but the question of how best to use the additional compute remains open. A popular approach is BoN sampling, where N candidate responses are generated, scored according to a reward model, and the highest-scoring response is selected. While this approach can improve performance, it is vulnerable to reward hacking, where performance degrades as N increases due to the selection of responses that exploit imperfections in the reward model instead of genuinely improving generation quality. Prior attempts to mitigate reward hacking, via stronger reward models or heavy-handed distributional regularization, either fail to fully address over-optimization or are too conservative to exploit additional compute. In this work, we explore the principle of pessimism in RL, which uses lower confidence bounds on value estimates to avoid OOD actions with uncertain reward estimates. Our approach, termed as caution, can be seen as the reverse of curiosity: where curiosity rewards prediction error as a signal of novelty, caution penalizes prediction error as a signal of distributional uncertainty. Practically, caution trains an error model on typical responses and uses its prediction error to lower reward estimates for atypical ones. Our extensive empirical evaluation demonstrates that caution is a simple, computationally efficient approach that substantially mitigates reward hacking in BoN sampling. We also provide a theoretical analysis in a simplified linear setting, which shows that caution provably improves over the standard BoN approach. Together, our results not only establish caution as a practical solution to reward hacking, but also provide evidence that curiosity-based approaches can be a general OOD detection technique in LLM settings.