# Learning U-Statistics with Active Inference

arXiv: https://arxiv.org/abs/2605.11638 · HTML: https://mlsanity.com/paper/2605.11638/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 53 references that are in the map.
- First posted on arXiv 12 May 2026.

## Builds on (most cited first, 23 of 23)

- [A Survey of Deep Active Learning](https://mlsanity.com/paper/2009.00236/) (2020)
- [A Theoretical Analysis of Contrastive Unsupervised Representation Learning](https://mlsanity.com/paper/1902.09229/) (2019)
- [Retiring Adult: New Datasets for Fair Machine Learning](https://mlsanity.com/paper/2108.04884/) (2021)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Prediction-Powered Inference](https://mlsanity.com/paper/2301.09633/) (2023)
- [PPI++: Efficient Prediction-Powered Inference](https://mlsanity.com/paper/2311.01453/) (2023)
- [Deep Active Learning: Unified and Principled Method for Query and Training](https://mlsanity.com/paper/1911.09162/) (2019)
- [Cross-Prediction-Powered Inference](https://mlsanity.com/paper/2309.16598/) (2023)
- [Active Statistical Inference](https://mlsanity.com/paper/2403.03208/) (2024)
- [Can Unconfident LLM Annotations Be Used for Confident Conclusions?](https://mlsanity.com/paper/2408.15204/) (2024)
- [A Statistical Perspective on Algorithmic Leveraging](https://mlsanity.com/paper/1306.5362/) (2013)
- [Semi-supervised Inference: General Theory and Estimation of Means](https://mlsanity.com/paper/1606.07268/) (2016)
- [Optimal Subsampling for Large Sample Logistic Regression](https://mlsanity.com/paper/1702.01166/) (2017)
- [Prediction-Powered Ranking of Large Language Models](https://mlsanity.com/paper/2402.17826/) (2024)
- [Prediction-Powered Inference with Imputed Covariates and Nonuniform Sampling](https://mlsanity.com/paper/2501.18577/) (2025)
- [Cost-Optimal Active AI Model Evaluation](https://mlsanity.com/paper/2506.07949/) (2025)
- [Doubly Robust Self-Training](https://mlsanity.com/paper/2306.00265/) (2023)
- [Scaling-up Empirical Risk Minimization: Optimization of Incomplete U-statistics](https://mlsanity.com/paper/1501.02629/) (2015)
- [Randomized incomplete U-statistics in high dimensions](https://mlsanity.com/paper/1712.00771/) (2017)
- [Robust Sampling for Active Statistical Inference](https://mlsanity.com/paper/2511.08991/) (2025)
- [Semi-Supervised U-statistics](https://mlsanity.com/paper/2402.18921/) (2024)
- [Inverse Probability Weighting: from Survey Sampling to Evidence Estimation](https://mlsanity.com/paper/2204.14121/) (2022)
- [On Differentially Private U Statistics](https://mlsanity.com/paper/2407.04945/) (2024)

## Abstract (arXiv)

U-statistics play a central role in statistical inference. In many modern applications, however, acquiring the labels required for U-statistics is costly. Motivated by recent advances in active inference, we develop an active inference framework for U-statistics that selectively queries informative labels to improve estimation efficiency under a fixed labeling budget, while preserving valid statistical inference. Our approach is built on the augmented inverse probability weighting U-statistic, which is designed to incorporate the sampling rule and machine learning predictions. We characterize the optimal sampling rule that minimizes its variance and design practical sampling strategies. We further extend the framework to U-statistic-based empirical risk minimization. Experiments on real datasets demonstrate substantial gains in estimation efficiency over baseline methods, while maintaining target coverage.