Differentially Private ℓ1-norm Linear Regression with Heavy-tailed Data

Abstract

We study the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) with heavy-tailed data. Specifically, we focus on the ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> -norm linear regression in the ϵ-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 ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> 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 $\tilde O\left( {\sqrt {\frac{d}{{n\varepsilon }}} } \right)$ (with high probability). Next, we relax the assumption to bounded θ-th order moment with some θ ∈ (1,2) and show that it is possible to achieve an upper bound of $\tilde O\left( {{{\left( {\sqrt {\frac{d}{{n\varepsilon }}} } \right)}^{\frac{{\theta - 1}}{\theta }}}} \right)$. 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 $\tilde O\left( {\sqrt {\frac{d}{{n\varepsilon }}} } \right)$ and $\tilde O\left( {\frac{d}{{{{\left( {n\varepsilon } \right)}^{\frac{{\theta - 1}}{\theta }}}}}} \right)$ in the second and θ-th moment case respectively.