High-Dimensional Private Empirical Risk Minimization by Greedy Coordinate Descent

TL;DR

This paper introduces a differentially private greedy coordinate descent algorithm that significantly reduces the impact of high dimensionality on utility in private empirical risk minimization, especially for structured problems.

Contribution

It proposes a novel DP-GCD algorithm that exploits problem structure to achieve logarithmic dimension dependence, improving privacy-utility trade-offs in high-dimensional settings.

Findings

DP-GCD achieves logarithmic dimension dependence in utility.

The algorithm performs well on synthetic and real datasets.

It exploits structural properties like quasi-sparsity for improved results.

Abstract

In this paper, we study differentially private empirical risk minimization (DP-ERM). It has been shown that the worst-case utility of DP-ERM reduces polynomially as the dimension increases. This is a major obstacle to privately learning large machine learning models. In high dimension, it is common for some model's parameters to carry more information than others. To exploit this, we propose a differentially private greedy coordinate descent (DP-GCD) algorithm. At each iteration, DP-GCD privately performs a coordinate-wise gradient step along the gradients' (approximately) greatest entry. We show theoretically that DP-GCD can achieve a logarithmic dependence on the dimension for a wide range of problems by naturally exploiting their structural properties (such as quasi-sparse solutions). We illustrate this behavior numerically, both on synthetic and real datasets.