# Differentially Private Nonparametric Modal Learning with Applications to Regression and Clustering

arXiv: https://arxiv.org/abs/2607.29675 · HTML: https://mlsanity.com/paper/2607.29675/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 67 references that are in the map.
- First posted on arXiv 31 Jul 2026.

## Builds on (most cited first, 9 of 9)

- [Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds](https://mlsanity.com/paper/1405.7085/) (2014)
- [Revisiting differentially private linear regression: optimal and adaptive prediction & estimation in unbounded domain](https://mlsanity.com/paper/1803.02596/) (2018)
- [Diffprivlib: The IBM Differential Privacy Library](https://mlsanity.com/paper/1907.02444/) (2019)
- [Differentially Private Simple Linear Regression](https://mlsanity.com/paper/2007.05157/) (2020)
- [Optimal Federated Learning for Nonparametric Regression with Heterogeneous Distributed Differential Privacy Constraints](https://mlsanity.com/paper/2406.06755/) (2024)
- [Stein's unbiased risk estimate and Hyv"arinen's score matching](https://mlsanity.com/paper/2502.20123/) (2025)
- [Differentially Private Kernel Density Estimation](https://mlsanity.com/paper/2409.01688/) (2024)
- [Effective Deterministic Initialization for k-Means-Like Methods via Local Density Peaks Searching](https://mlsanity.com/paper/1611.06777/) (2016)
- [Pointwise adaptive kernel density estimation under local approximate differential privacy](https://mlsanity.com/paper/1907.06233/) (2019)

## Abstract (arXiv)

Density modes provide a localized and interpretable summary of multimodal distributions, but their estimation under rigorous differential privacy constraints remains largely unexplored. We study differentially private recovery of density modes for multivariate distributions under local smoothness, curvature, and separation conditions. We propose DP-GRAMS, a mean-shift inspired method that performs noisy ascent on a differentially private score estimator. Assuming the density belongs locally to a H"older class with smoothness parameter β > 2, our score estimator uses bias-reducing higher-order kernels, and then enforces privacy in the gradient ascent steps via gradient clipping and calibrated Gaussian noise. A private initialization scheme combines a density-aware utility with a suppression rule and, with kasymp Mlog n draws over a public hDAP-grid and suppression radius ρinitasymp (log n)-1/d, achieves high-probability coverage of the modal basins by successively suppressing selected local neighborhoods in competitive regions, while correlated noise across multiple starts enables joint release under a single (varepsilon,δ)-differential privacy guarantee. We prove that all population modes are recovered with high probability and establish asymptotic error rates of the form O!left((tfraclog nn)^frac2(β-1)d+2βright) + O!left((tfracpolylog(n,δ)n2varepsilon2)^fracβ-1d+βright). We also provide minimax lower bounds for private mode estimation, and show that our estimators are nearly optimal, up to a logarithmic factor in the MSE. We present two natural extensions: DP-PMS, a private modal-regression method, and DP-GRAMS-C, a clustering pipeline. Extensive experiments on synthetic and real data demonstrate favorable privacy-utility trade-offs relative to common baselines.