Differentially Private $\ell_1$-norm Linear Regression with Heavy-tailed Data

TL;DR

This paper develops differentially private algorithms for $ ext{l}_1$-norm linear regression with heavy-tailed data, achieving near-optimal error bounds under relaxed moment assumptions.

Contribution

It introduces new DP algorithms for heavy-tailed data in $ ext{l}_1$-norm regression, extending prior work to bounded moment conditions beyond Lipschitz loss functions.

Findings

Achieves $ ilde{O}(rac{ ext{d}}{ ext{n} ext{epsilon}})^{1/2}$ error under second moment assumption.

Extends to $ heta$-th moment with error $ ilde{O}((rac{ ext{d}}{ ext{n} ext{epsilon}})^{( heta-1)/ heta})$.

Handles coordinate-wise bounded moments with similar bounds.

Abstract

We study the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) with heavy-tailed data. Specifically, we focus on the $\ell_1$-norm linear regression in the $\epsilon$-DP model. While most of the previous work focuses on the case where the loss function is Lipschitz, here we only need to assume the variates has bounded moments. Firstly, we study the case where the $\ell_2$ norm of data has bounded second order moment. We propose an algorithm which is based on the exponential mechanism and show that it is possible to achieve an upper bound of $\tilde{O}(\sqrt{\frac{d}{n\epsilon}})$ (with high probability). Next, we relax the assumption to bounded $\theta$-th order moment with some $\theta\in (1, 2)$ and show that it is possible to achieve an upper bound of $\tilde{O}(({\frac{d}{n\epsilon}})^\frac{\theta-1}{\theta})$. Our algorithms can also be extended to more relaxed cases where only each coordinate of the data has bounded moments, and we can get an upper bound of $\tilde{O}({\frac{d}{\sqrt{n\epsilon}}})$ and $\tilde{O}({\frac{d}{({n\epsilon})^\frac{\theta-1}{\theta}}})$ in the second and $\theta$-th moment case respectively.