Best-of-N (BoN) sampling with a reward model has been shown to be an effective strategy for aligning Large Language Models (LLMs) with human preferences at the time of decoding. BoN sampling is susceptible to a problem known as reward hacking. Since the reward model is an imperfect proxy for the true objective, an excessive focus on optimizing its value can lead to a compromise of its performance on the true objective. Previous work proposes Regularized BoN sampling (RBoN), a BoN sampling with regularization to the objective, and shows that it outperforms BoN sampling so that it mitigates reward hacking and empirically (Jinnai et al., 2024). However, Jinnai et al. (2024) introduce RBoN based on a heuristic and they lack the analysis of why such regularization strategy improves the performance of BoN sampling. The aim of this study is to analyze the effect of BoN sampling on regularization strategies. Using the regularization strategies corresponds to robust optimization, which maximizes the worst case over a set of possible perturbations in the proxy reward. Although the theoretical guarantees are not directly applicable to RBoN, RBoN corresponds to a practical implementation. This paper proposes an extension of the RBoN framework, called Stochastic RBoN sampling (SRBoN), which is a theoretically guaranteed approach to worst-case RBoN in proxy reward. We then perform an empirical evaluation using the AlpacaFarm and Anthropic's hh-rlhf datasets to evaluate which factors of the regularization strategies contribute to the improvement of the true proxy reward. In addition, we also propose another simple RBoN method, the Sentence Length Regularized BoN, which has a better performance in the experiment as compared to the previous methods.

Best-of-N (BoN) sampling with a reward model has been shown to be an effective strategy for aligning Large Language Models (LLMs) to human preferences at the time of decoding. BoN sampling is susceptible to a problem known as reward hacking. Because the reward model is an imperfect proxy for the true objective, over-optimizing its value can compromise its performance on the true objective. A common solution to prevent reward hacking in preference learning techniques is to optimize a reward using proximity regularization (e.g., KL regularization), which ensures that the language model remains close to the reference model. In this research, we propose Regularized Best-of-N (RBoN), a variant of BoN that aims to mitigate reward hacking by incorporating a proximity term in response selection, similar to preference learning techniques. We evaluate RBoN on the AlpacaFarm and Anthropic's hh-rlhf datasets and find that it outperforms BoN. As an application of RBoN, we use RBoN to generate a pairwise preference learning dataset. Experimental results show that a DPO model trained on a dataset generated with RBoN outperforms a DPO model generated with vanilla BoN. Our code is available at https://github.com/CyberAgentAILab/regularized-bon