Faster Differentially Private Convex Optimization via Second-Order Methods

TL;DR

This paper introduces a differentially private second-order optimization method that accelerates convex learning, achieving faster convergence and better accuracy than existing first-order DP algorithms, especially in logistic regression.

Contribution

The paper develops a private second-order Newton-based algorithm with quadratic convergence for strongly convex functions, improving efficiency and accuracy in DP convex optimization.

Findings

Achieves quadratic convergence for strongly convex functions.

Outperforms DP-GD/DP-SGD by 10-40x in speed.

Consistently yields lower excess loss in experiments.

Abstract

Differentially private (stochastic) gradient descent is the workhorse of DP private machine learning in both the convex and non-convex settings. Without privacy constraints, second-order methods, like Newton's method, converge faster than first-order methods like gradient descent. In this work, we investigate the prospect of using the second-order information from the loss function to accelerate DP convex optimization. We first develop a private variant of the regularized cubic Newton method of Nesterov and Polyak, and show that for the class of strongly convex loss functions, our algorithm has quadratic convergence and achieves the optimal excess loss. We then design a practical second-order DP algorithm for the unconstrained logistic regression problem. We theoretically and empirically study the performance of our algorithm. Empirical results show our algorithm consistently achieves the best excess loss compared to other baselines and is 10-40x faster than DP-GD/DP-SGD.