What should post-training optimize? A test-time scaling law perspective
For practitioners deploying LLMs with test-time compute scaling, this work provides a practical post-training method that bridges the gap between limited training budgets and large deployment budgets, improving best-of-N performance without requiring expensive training rollouts.
The paper addresses the mismatch between standard post-training objectives (optimizing mean reward of a single response) and best-of-N deployment performance (governed by upper tail of reward distribution), especially when training has limited per-prompt rollouts (m << N). They propose Tail-Extrapolated estimators (TEA and Prefix-TEA) that approximate the best-of-N objective from small rollout groups, achieving improved best-of-N performance across various models and datasets.
Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.