# Refined Differentially Private Linear Regression via Extension of a Free Lunch Result

arXiv: https://arxiv.org/abs/2604.11820 · HTML: https://mlsanity.com/paper/2604.11820/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 20 references that are in the map.
- First posted on arXiv 11 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Concentrated Differential Privacy: Simplifications, Extensions, and Lower Bounds](https://mlsanity.com/paper/1605.02065/) (2016)
- [Private Mean Estimation of Heavy-Tailed Distributions](https://mlsanity.com/paper/2002.09464/) (2020)
- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [Average-Case Averages: Private Algorithms for Smooth Sensitivity and Mean Estimation](https://mlsanity.com/paper/1906.02830/) (2019)
- [Differentially Private Simple Linear Regression](https://mlsanity.com/paper/2007.05157/) (2020)
- [How Private are DP-SGD Implementations?](https://mlsanity.com/paper/2403.17673/) (2024)
- [Easy Differentially Private Linear Regression](https://mlsanity.com/paper/2208.07353/) (2022)
- [Mean estimation in the add-remove model of differential privacy](https://mlsanity.com/paper/2312.06658/) (2023)
- [To Shuffle or not to Shuffle: Auditing DP-SGD with Shuffling](https://mlsanity.com/paper/2411.10614/) (2024)
- [Private Regression via Data-Dependent Sufficient Statistic Perturbation](https://mlsanity.com/paper/2405.15002/) (2024)
- [Private Means and the Curious Incident of the Free Lunch](https://mlsanity.com/paper/2408.10438/) (2024)
- [Differentially Private Linear Regression and Synthetic Data Generation with Statistical Guarantees](https://mlsanity.com/paper/2510.16974/) (2025)

## Abstract (arXiv)

As data-privacy regulations tighten and statistical models are increasingly deployed on sensitive human-sourced data, privacy-preserving linear regression has become a critical necessity. For the add-remove DP model, Kulesza et al. (2024) and Fitzsimons et al. (2024) have independently shown that the size of the dataset -- an important statistic for linear regression -- can be privately estimated for "free", via a simplex transformation of bounded variables and private sum queries on the transformed variables. In this work, we extend this free lunch result via carefully crafted multidimensional simplex transformations to variables and functions that are bounded in the interval [0,1]. We show that these transformations can be applied to refine the estimates of sufficient statistics needed for private simple linear regression based on ordinary least squares. We provide both analytical and numerical results to demonstrate the superiority of our approach. Our proposed transformations have general applicability and can be readily adapted for differentially private polynomial regression.