# Agnostic Private Density Estimation for GMMs via List Global Stability

arXiv: https://arxiv.org/abs/2407.04783 · HTML: https://mlsanity.com/paper/2407.04783/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 62 references that are in the map.
- First posted on arXiv 5 Jul 2024.

## Builds on (most cited first, 30 of 43)

- [Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures](https://mlsanity.com/paper/1611.03473/) (2016)
- [Finite Sample Differentially Private Confidence Intervals](https://mlsanity.com/paper/1711.03908/) (2017)
- [Privately Learning High-Dimensional Distributions](https://mlsanity.com/paper/1805.00216/) (2018)
- [Fingerprinting Codes and the Price of Approximate Differential Privacy](https://mlsanity.com/paper/1311.3158/) (2013)
- [CoinPress: Practical Private Mean and Covariance Estimation](https://mlsanity.com/paper/2006.06618/) (2020)
- [Robust and Differentially Private Mean Estimation](https://mlsanity.com/paper/2102.09159/) (2021)
- [An Equivalence Between Private Classification and Online Prediction](https://mlsanity.com/paper/2003.00563/) (2020)
- [Differential privacy and robust statistics in high dimensions](https://mlsanity.com/paper/2111.06578/) (2021)
- [Private Learning and Sanitization: Pure vs. Approximate Differential Privacy](https://mlsanity.com/paper/1407.2674/) (2014)
- [Differentially Private Release and Learning of Threshold Functions](https://mlsanity.com/paper/1504.07553/) (2015)
- [Private Hypothesis Selection](https://mlsanity.com/paper/1905.13229/) (2019)
- [Efficient Mean Estimation with Pure Differential Privacy via a Sum-of-Squares Exponential Mechanism](https://mlsanity.com/paper/2111.12981/) (2021)
- [Robustness Implies Privacy in Statistical Estimation](https://mlsanity.com/paper/2212.05015/) (2022)
- [Reproducibility in Learning](https://mlsanity.com/paper/2201.08430/) (2022)
- [Stability is Stable: Connections between Replicability, Privacy, and Adaptive Generalization](https://mlsanity.com/paper/2303.12921/) (2023)
- [On the Sample Complexity of Privately Learning Unbounded High-Dimensional Gaussians](https://mlsanity.com/paper/2010.09929/) (2020)
- [Private and polynomial time algorithms for learning Gaussians and beyond](https://mlsanity.com/paper/2111.11320/) (2021)
- [Private Robust Estimation by Stabilizing Convex Relaxations](https://mlsanity.com/paper/2112.03548/) (2021)
- [Simultaneous Private Learning of Multiple Concepts](https://mlsanity.com/paper/1511.08552/) (2015)
- [Differentially Private Clustering: Tight Approximation Ratios](https://mlsanity.com/paper/2008.08007/) (2020)
- [A Private and Computationally-Efficient Estimator for Unbounded Gaussians](https://mlsanity.com/paper/2111.04609/) (2021)
- [New Lower Bounds for Private Estimation and a Generalized Fingerprinting Lemma](https://mlsanity.com/paper/2205.08532/) (2022)
- [Private Estimation with Public Data](https://mlsanity.com/paper/2208.07984/) (2022)
- [Sample-efficient proper PAC learning with approximate differential privacy](https://mlsanity.com/paper/2012.03893/) (2020)
- [Differentially Private Algorithms for Learning Mixtures of Separated Gaussians](https://mlsanity.com/paper/1909.03951/) (2019)
- [From Robustness to Privacy and Back](https://mlsanity.com/paper/2302.01855/) (2023)
- [Realizable Learning is All You Need](https://mlsanity.com/paper/2111.04746/) (2021)
- [Privacy Induces Robustness: Information-Computation Gaps and Sparse Mean Estimation](https://mlsanity.com/paper/2211.00724/) (2022)
- [Private estimation algorithms for stochastic block models and mixture models](https://mlsanity.com/paper/2301.04822/) (2023)
- [Polynomial Time and Private Learning of Unbounded Gaussian Mixture Models](https://mlsanity.com/paper/2303.04288/) (2023)

## Built on by (most cited first, 2 of 2)

- [Sample-Efficient Private Learning of Mixtures of Gaussians](https://mlsanity.com/paper/2411.02298/) (2024)
- [Canonicalized Stable-List Replay for Private Federated Continual Learning over Language-Model Embeddings](https://mlsanity.com/paper/2606.00426/) (2026)

## Abstract (arXiv)

We consider the problem of private density estimation for mixtures of unrestricted high dimensional Gaussians in the agnostic setting. We prove the first upper bound on the sample complexity of this problem. Previously, private learnability of high dimensional GMMs was only known in the realizable setting [Afzali et al., 2024]. To prove our result, we exploit the notion of textitlist global stability [Ghazi et al., 2021b,a] that was originally introduced in the context of private supervised learning. We define an agnostic variant of this definition, showing that its existence is sufficient for agnostic private density estimation. We then construct an agnostic list globally stable learner for GMMs.