Optimal differentially private kernel learning with random projection

TL;DR

This paper introduces a novel differentially private kernel learning algorithm using random projection, achieving optimal excess risk bounds and outperforming existing methods in privacy-utility trade-offs.

Contribution

It presents a new private kernel ERM algorithm based on random projection that attains minimax-optimal risk rates and offers the first dimension-free excess risk bounds for private linear ERM.

Findings

Achieves minimax-optimal excess risk rates for squared and Lipschitz-smooth convex losses.

Demonstrates that random projection outperforms random Fourier features and $ ext{l}_2$ regularization.

Provides empirical evidence supporting the efficiency and optimality of the proposed method.

Abstract

Differential privacy has become a cornerstone in the development of privacy-preserving learning algorithms. This work addresses optimizing differentially private kernel learning within the empirical risk minimization (ERM) framework. We propose a novel differentially private kernel ERM algorithm based on random projection in the reproducing kernel Hilbert space using Gaussian processes. Our method achieves minimax-optimal excess risk rates for both the squared loss and Lipschitz-smooth convex loss functions under a local strong convexity condition. We further show that existing approaches based on alternative dimension reduction techniques, such as random Fourier feature mappings or $\ell_2$ regularization, yield suboptimal excess risk bounds. Our key theoretical contribution also includes the derivation of dimension-free excess risk bounds for objective perturbation-based private linear ERM, marking the first such result that does not rely on noisy gradient-based mechanisms. Additionally, we obtain sharper excess risk bounds for existing differentially private kernel ERM algorithms. Empirical evaluations support our theoretical claims, demonstrating that random projection enables statistically efficient and optimally private kernel learning. These findings provide new insights into the design of differentially private algorithms and highlight the central role of dimension reduction in balancing privacy and utility.