(Nearly) Optimal Private Linear Regression for Sub-Gaussian Data via Adaptive Clipping

Abstract

We study the problem of differentially private linear regression where each of the data point is sampled from a fixed sub-Gaussian style distribution. We propose and analyze a one-pass mini-batch stochastic gradient descent method (DP-AMBSSGD) where points in each iteration are sampled without replacement. Noise is added for DP but the noise standard deviation is estimated online. Compared to existing $(\epsilon, \delta)$-DP techniques which have sub-optimal error bounds, DP-AMBSSGD is able to provide nearly optimal error bounds in terms of key parameters like dimensionality $d$, number of points $N$, and the standard deviation \sigma of the noise in observations. For example, when the $d$-dimensional covariates are sampled i.i.d. from the normal distribution, then the excess error of DP-AMBSSGD due to privacy is $\sigma^2 d/N(1+d/(\epsilon^2 N))$, i.e., the error is meaningful when number of samples N\geq d \log d which is the standard operative regime for linear regression. In contrast, error bounds for existing efficient methods in this setting are: $d^3/(\epsilon^2 N^2)$, even for $\sigma=0$. That is, for constant $\epsilon$, the existing techniques require $N=d^{1.5}$ to provide a non-trivial result.