Theoretical guarantees on the best-of-n alignment policy

TL;DR

This paper critically examines the theoretical properties of best-of-$n$ sampling in generative models, correcting a common misconception about KL divergence bounds, and introduces a new estimator to better understand the tradeoffs involved.

Contribution

It disproves a widely cited analytical expression for KL divergence in best-of-$n$ sampling, proposes a new estimator, and analyzes the tradeoffs between win rate and divergence.

Findings

The claimed KL divergence formula is an upper bound, not an exact value.

A new KL divergence estimator provides a tighter approximation.

Good tradeoffs between win rate and divergence are achievable with less than 1000 samples.

Abstract

A simple and effective method for the inference-time alignment and scaling test-time compute of generative models is best-of-$n$ sampling, where $n$ samples are drawn from a reference policy, 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 reference 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, and propose a new estimator for the KL divergence and empirically show that it provides a tight approximation. We also show that the win rate of the best-of-$n$ policy against the reference policy is upper bounded by $n/(n+1)$ and derive bounds on the tightness of this characterization. We conclude with analyzing the tradeoffs between win rate and KL divergence of the best-of-$n$ alignment policy, which demonstrate that very good tradeoffs are achievable with $n < 1000$.