Instance-Optimal Estimation with Multiple LLM Judges on a Budget

This paper addresses the cost-efficient evaluation of large language models (LLMs) by utilizing multiple AI "judges" with different price points and reliability levels. The researchers formalize this challenge as budgeted heteroskedastic multi-judge estimation, seeking an optimal way to distribute a limited budget across various judges and tasks to achieve the most accurate quality scores. They introduce EST-IVWE, an adaptive algorithm that learns the unknown variances of different judges and assigns resources to those providing the best cost-to-variance trade-off. Through rigorous proofs, the authors demonstrate that their approach is instance-optimal, meaning it achieves the best possible accuracy for any specific set of judges and prompts. Furthermore, the paper provides a theoretical breakthrough by showing that specialized mathematical arguments are required to capture the true geometric structure of this allocation problem. Numerical experiments on synthetic and real-world datasets confirm that this adaptive strategy significantly outperforms simple uniform budgeting.