# The Cost of Privacy in Generalized Linear Models: Algorithms and Minimax Lower Bounds

arXiv: https://arxiv.org/abs/2011.03900 · HTML: https://mlsanity.com/paper/2011.03900/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2020.

## Built on by (most cited first, 15 of 15)

- [High Dimensional Differentially Private Stochastic Optimization with Heavy-tailed Data](https://mlsanity.com/paper/2107.11136/) (2021)
- [New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma](https://mlsanity.com/paper/2205.08532/) (2022)
- [Differentially private inference via noisy optimization](https://mlsanity.com/paper/2103.11003/) (2021)
- [Faster Rates of Private Stochastic Convex Optimization](https://mlsanity.com/paper/2108.00331/) (2021)
- [Fast Composite Optimization and Statistical Recovery in Federated Learning](https://mlsanity.com/paper/2207.08204/) (2022)
- [A Bias-Accuracy-Privacy Trilemma for Statistical Estimation](https://mlsanity.com/paper/2301.13334/) (2023)
- [Differentially Private Regression with Unbounded Covariates](https://mlsanity.com/paper/2202.11199/) (2022)
- [A Central Limit Theorem for Differentially Private Query Answering](https://mlsanity.com/paper/2103.08721/) (2021)
- [High-Dimensional Differentially-Private EM Algorithm: Methods and Near-Optimal Statistical Guarantees](https://mlsanity.com/paper/2104.00245/) (2021)
- [Differentially Private Optimization with Sparse Gradients](https://mlsanity.com/paper/2404.10881/) (2024)
- [Differentially Private Federated Learning: Servers Trustworthiness, Estimation, and Statistical Inference](https://mlsanity.com/paper/2404.16287/) (2024)
- [Truthful Generalized Linear Models](https://mlsanity.com/paper/2209.07815/) (2022)
- [Differentially Private Sparse Linear Regression with Heavy-tailed Responses](https://mlsanity.com/paper/2506.06861/) (2025)
- [Optimal Differentially Private Ranking from Pairwise Comparisons](https://mlsanity.com/paper/2507.09388/) (2025)
- [Differentially Private High Dimensional Bandits](https://mlsanity.com/paper/2402.03737/) (2024)

## Abstract (arXiv)

We propose differentially private algorithms for parameter estimation in both low-dimensional and high-dimensional sparse generalized linear models (GLMs) by constructing private versions of projected gradient descent. We show that the proposed algorithms are nearly rate-optimal by characterizing their statistical performance and establishing privacy-constrained minimax lower bounds for GLMs. The lower bounds are obtained via a novel technique, which is based on Stein's Lemma and generalizes the tracing attack technique for privacy-constrained lower bounds. This lower bound argument can be of independent interest as it is applicable to general parametric models. Simulated and real data experiments are conducted to demonstrate the numerical performance of our algorithms.