No Free Lunch: Non-Asymptotic Analysis of Prediction-Powered Inference

2025-05-26

AI Summary

This paper investigates the statistical performance of Prediction-Driven Inference (PPI++) under limited-sample regimes, specifically characterizing when its estimation error degrades relative to inference using only ground-truth labels. Through non-asymptotic analysis and a correlation-driven error decomposition, we derive the first exact finite-sample error bound for PPI++. We establish a “no-free-lunch” theorem: PPI++ improves estimation accuracy if and only if the correlation between pseudo-labels and ground-truth labels exceeds $1/sqrt{n-2}$ in the Gaussian setting—a threshold rigorously derived for both binary and Gaussian cases. We further analyze trade-offs between single-sample and sample-splitting variants of PPI++. All theoretical findings are empirically validated on real-world datasets, confirming that the predicted correlation threshold accurately governs performance gains.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Humans versus LLMs for data annotation and labeling
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Prediction-Powered Inference (PPI) is a popular strategy for combining gold-standard and possibly noisy pseudo-labels to perform statistical estimation. Prior work has shown an asymptotic"free lunch"for PPI++, an adaptive form of PPI, showing that the *asymptotic* variance of PPI++ is always less than or equal to the variance obtained from using gold-standard labels alone. Notably, this result holds *regardless of the quality of the pseudo-labels*. In this work, we demystify this result by conducting an exact finite-sample analysis of the estimation error of PPI++ on the mean estimation problem. We give a"no free lunch"result, characterizing the settings (and sample sizes) where PPI++ has provably worse estimation error than using gold-standard labels alone. Specifically, PPI++ will outperform if and only if the correlation between pseudo- and gold-standard is above a certain level that depends on the number of labeled samples ($n$). In some cases our results simplify considerably: For Gaussian data, the correlation must be at least $1/sqrt{n - 2}$ in order to see improvement, and a similar result holds for binary labels. In experiments, we illustrate that our theoretical findings hold on real-world datasets, and give insights into trade-offs between single-sample and sample-splitting variants of PPI++.

Problem

• Analyzes finite-sample error of PPI++ mean estimation
• Identifies conditions where PPI++ underperforms gold-standard labels
• Determines minimum pseudo-label correlation for PPI++ improvement

Innovation

• Exact finite-sample analysis of PPI++
• Correlation threshold determines PPI++ performance
• Gaussian and binary label specific results