这篇论文的DP-GRAMS能在差分隐私下找到密度模式，误差接近理论最优，还附赠回归和聚类版本。
论文提出DP-GRAMS，一种基于均值漂移的差分隐私模态估计方法，用于多变量密度。该方法在Hölder平滑参数β>2时采用高阶核降低偏差，并用梯度裁剪和校准高斯噪声保护隐私。理论证明所有总体模态能以高概率恢复，渐近误差率为O((log n/n)^{2(β-1)/(d+2β)}) + O((polylog(n,δ)/(n^2ε^2))^{(β-1)/(d+β)})。作者还给出私有模态估计的极小极大下界，表明该估计器在均方误差上近乎最优，仅差对数因子。扩展版本DP-PMS和DP-GRAMS-C分别用于回归和聚类，实验显示隐私-效用权衡优于常见基线。
Differentially Private Nonparametric Modal Learning with Applications to Regression and Clustering
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 Hölder 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 $k\asymp M\log n$ draws over a public $h_{\mathrm{DAP}}$-grid and suppression radius $ρ_{\mathrm{init}}\asymp (\log n)^{-1/d}$, 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 $O\!\left((\tfrac{\log n}{n})^{\frac{2(β-1)}{d+2β}}\right) + O\!\left((\tfrac{\mathrm{polylog}(n,δ)}{n^2\varepsilon^2})^{\frac{β-1}{d+β}}\right)$. 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.