# Scale-invariant Optimal Sampling for Rare-events Data with Sparse Models

arXiv: https://arxiv.org/abs/2608.22597 · HTML: https://mlsanity.com/paper/2608.22597/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 30 references that are in the map.
- First posted on arXiv 23 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Optimal Subsampling for Large Sample Logistic Regression](https://mlsanity.com/paper/1702.01166/) (2017)
- [Optimal Distributed Subsampling for Maximum Quasi-Likelihood Estimators with Massive Data](https://mlsanity.com/paper/2005.10435/) (2020)
- [Optimal Subsampling Algorithms for Big Data Regressions](https://mlsanity.com/paper/1806.06761/) (2018)
- [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)
- [Nonuniform Negative Sampling and Log Odds Correction with Rare Events Data](https://mlsanity.com/paper/2110.13048/) (2021)
- [Optimal Cox Regression Subsampling Procedure with Rare Events](https://mlsanity.com/paper/2012.02122/) (2020)
- [Optimal Sampling for Generalized Linear Models under Measurement Constraints](https://mlsanity.com/paper/1907.07309/) (2019)

## Abstract (arXiv)

Subsampling is effective in tackling computational challenges for massive data with rare events. Overly aggressive subsampling may adversely affect estimation efficiency, and optimal subsampling is essential to mitigate the information loss. However, existing optimal subsampling probabilities depend on data scales, and some scaling transformations may result in inefficient subsamples. This problem is more significant when there are inactive features, because their influence on the subsampling probabilities can be arbitrarily magnified by inappropriate scaling transformations. We tackle this challenge and introduce a scale-invariant optimal subsampling function in the context of sparse models, where inactive features are commonly assumed. Instead of focusing on estimating model parameters, we define an optimal subsampling function to minimize the prediction error, using adaptive lasso to outline the estimation procedure and study its theoretical guarantee. We first introduce the adaptive lasso estimator for rare-events data and establish its oracle properties, thereby validating the use of subsampling. Then we derive a scale-invariant optimal subsampling function that minimizes the prediction error of the inverse probability weighted (IPW) adaptive lasso. Finally, we present an estimator based on the maximum sampled conditional likelihood (MSCL) to further improve the estimation efficiency. We conduct numerical experiments using both simulated and real-world data sets to demonstrate the performance of the proposed methods.