Differentially Private Nonparametric Modal Learning with Applications to Regression and Clustering
Authors: Arkajyoti Bhattacharjee, Arnab Auddy
Organizations: Department of Statistics The Ohio State University
Abstract
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≍Mlogn draws over a public hDAP-grid and suppression radius ρinit≍(logn)−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 (ε,δ)-differential privacy guarantee. We prove that all population modes are recovered with high probability and establish asymptotic error rates of the form O((nlogn)d+2β2(β−1))+O((n2ε2polylog(n,δ))d+ββ−1). 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.
Differential privacy (DP) has become the gold standard for ensuring the privacy protection of machine learning and statistical algorithms in recent decades. A plethora of algorithms and methods have been developed to enhance the utility of DP algorithms while maintaining the same level of DP. However, these are often overly complex or computationally ineffective. We propose a novel approach focusing on denoising the output of the simple additive Gaussian mechanism by adopting the idea of \textit{empirical Bayes estimation}. We highlight that the empirical Bayes approach can reduce the mean-squared error solely by taking the output of the Gaussian mechanism as input. Our numerical studies show that this simple yet powerful approach can be applied to improve upon various statistical problems, including histogram release, principal component analysis, and linear regression, often outperforming existing private algorithms.
In machine learning, privacy requirements at inference or deployment time often evolve due to changing policies, regulations, or user preferences. In this work, we aim to construct a magnitude of models to satisfy any target differential privacy (DP) requirement without additional training, given a set of existing models trained on the same dataset with different privacy/utility tradeoffs. We propose two post-processing techniques, namely random selection and linear combination, to generate final private models satisfying any target privacy parameter. We provide privacy accounting of these approaches from the lens of R'enyi DP and privacy loss distributions on general problems, as well as on private mean estimation, where we precisely characterize the privacy/utility tradeoffs and compare the two mechanisms. Empirically, we demonstrate the effectiveness of our approaches and validate our analyses on several models and both synthetic and real-world datasets.
We study differentially private (DP) regression in settings where each data sample includes public, non-sensitive features -- common in applications such as recommendation and advertising systems. While such label-DP or semi-sensitive-feature settings have been primarily explored in the context of classification, effective approaches for regression remain underexplored. We introduce Cond-DP, a conditioned variant of DPSGD that leverages the structure of public feature matrices to improve optimization under privacy constraints. Motivated by the observation that these public features often exhibit rapidly decaying spectra, Cond-DP incorporates a data-driven conditioning matrix to reshape the optimization landscape and accelerate convergence. We provide convergence guarantees for convex, strongly convex, and non-convex settings, and recover standard DPSGD as a special case when the conditioning matrix is the identity. We show how to construct an effective conditioning matrix for Cond-DP directly from public features, enabling provably faster convergence than DPSGD in private linear regression without incurring additional privacy cost. Empirically, Cond-DP with this conditioning matrix consistently outperforms state-of-the-art baselines across a wide range of datasets and model architectures under label DP, demonstrating strong and robust performance in practice.