A simple and effective method for the alignment of generative models is the best-of-$n$ policy, where $n$ samples are drawn from a base policy, and ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-$n$ policy and the base policy is equal to $\log (n) - (n-1)/n.$ We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes. Finally, we propose a new estimator for the KL divergence and empirically show that it provides a tight approximation through a few examples.

翻译：一种简单有效的生成模型对齐方法是最优n选策略（best-of-$n$ policy），即从基础策略中抽取$n$个样本，根据奖励函数排序并选择排名最高的样本。文献中常用的一种解析表达式声称，最优n选策略与基础策略之间的KL散度等于$\log (n) - (n-1)/n$。我们否定了该声明的有效性，并证明它实际上是真实KL散度的一个上界。我们还探讨了该上界在不同情况下的紧致性。最后，我们提出了一个新的KL散度估计器，并通过若干实例从经验上证明它能提供紧致的近似值。