Fine-Tune, Then Rectify

Driven by recent advances in artificial intelligence, a growing literature has demonstrated the potential of using large language models (LLMs) as scalable surrogates to generate human-like responses. Two common approaches to improve the performance of LLMs include: fine-tuning, which aligns the LLM more closely with human responses, and rectification, which corrects biases in LLM outputs. In this paper, we develop a two-stage framework that combines fine-tuning and rectification, and optimally allocates limited labeled samples across the two stages. A key insight is that the conventional fine-tuning objective of minimizing mean squared prediction error is generally not aligned with the downstream rectification stage. For mean estimation, we propose to minimize the variance of the prediction errors; for general M-estimation, we propose to minimize a scalarized variance metric as the fine-tuning objective. Building on this insight, we leverage the scaling law of fine-tuning to optimally allocate the limited labeled human data between the fine-tuning and rectification stages. Our empirical analysis validates the fine-tuning scaling law and confirms that our proposed optimal allocation rule reliably identifies the optimal sample allocation. We demonstrate substantial efficiency gains in estimation and inference performance relative to fine-tuning or rectification alone, or to employing the conventional mean squared error objective within the fine-tuning then rectification framework. Such efficiency gains translate to significant cost savings for making reliable decisions.