# Nonuniform Negative Sampling and Log Odds Correction with Rare Events Data

arXiv: https://arxiv.org/abs/2110.13048 · HTML: https://mlsanity.com/paper/2110.13048/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 49 references that are in the map.
- First posted on arXiv 25 Oct 2021.

## Builds on (most cited first, 12 of 12)

- [DeepFM: A Factorization-Machine based Neural Network for CTR Prediction](https://mlsanity.com/paper/1703.04247/) (2017)
- [Imbalanced-learn: A Python Toolbox to Tackle the Curse of Imbalanced Datasets in Machine Learning](https://mlsanity.com/paper/1609.06570/) (2016)
- [Embedding-based Retrieval in Facebook Search](https://mlsanity.com/paper/2006.11632/) (2020)
- [Deep CTR Prediction in Display Advertising](https://mlsanity.com/paper/1609.06018/) (2016)
- [Optimal Subsampling for Large Sample Logistic Regression](https://mlsanity.com/paper/1702.01166/) (2017)
- [Generalization in anti-causal learning](https://mlsanity.com/paper/1812.00524/) (2018)
- [Local case-control sampling: Efficient subsampling in imbalanced data sets](https://mlsanity.com/paper/1306.3706/) (2013)
- [Optimal Distributed Subsampling for Maximum Quasi-Likelihood Estimators with Massive Data](https://mlsanity.com/paper/2005.10435/) (2020)
- [Optimal subsampling for quantile regression in big data](https://mlsanity.com/paper/2001.10168/) (2020)
- [Logistic Regression for Massive Data with Rare Events](https://mlsanity.com/paper/2006.00683/) (2020)
- [More Efficient Estimation for Logistic Regression with Optimal Subsample](https://mlsanity.com/paper/1802.02698/) (2018)
- [Local Uncertainty Sampling for Large-Scale Multi-Class Logistic Regression](https://mlsanity.com/paper/1604.08098/) (2016)

## Built on by (most cited first, 4 of 4)

- [Graph-Based Model-Agnostic Data Subsampling for Recommendation Systems](https://mlsanity.com/paper/2305.16391/) (2023)
- [A Provably Accurate Randomized Sampling Algorithm for Logistic Regression](https://mlsanity.com/paper/2402.16326/) (2024)
- [Scalable Counterfactual Risk Estimation for Rare Events in Longitudinal Data](https://mlsanity.com/paper/2606.01539/) (2026)
- [Scale-invariant Optimal Sampling for Rare-events Data with Sparse Models](https://mlsanity.com/paper/2608.22597/) (2026)

## Abstract (arXiv)

We investigate the issue of parameter estimation with nonuniform negative sampling for imbalanced data. We first prove that, with imbalanced data, the available information about unknown parameters is only tied to the relatively small number of positive instances, which justifies the usage of negative sampling. However, if the negative instances are subsampled to the same level of the positive cases, there is information loss. To maintain more information, we derive the asymptotic distribution of a general inverse probability weighted (IPW) estimator and obtain the optimal sampling probability that minimizes its variance. To further improve the estimation efficiency over the IPW method, we propose a likelihood-based estimator by correcting log odds for the sampled data and prove that the improved estimator has the smallest asymptotic variance among a large class of estimators. It is also more robust to pilot misspecification. We validate our approach on simulated data as well as a real click-through rate dataset with more than 0.3 trillion instances, collected over a period of a month. Both theoretical and empirical results demonstrate the effectiveness of our method.