September 23, 2021

Dedicated Forum to help removing adware, malware, spyware, ransomware, trojans, viruses and more!

High Dimensional Differentially Private Stochastic Optimization with Heavy-tailed Data. (arXiv:2107.11136v1 [cs.LG])

As one of the most fundamental problems in machine learning, statistics and
differential privacy, Differentially Private Stochastic Convex Optimization
(DP-SCO) has been extensively studied in recent years. However, most of the
previous work can only handle either regular data distribution or irregular
data in the low dimensional space case. To better understand the challenges
arising from irregular data distribution, in this paper we provide the first
study on the problem of DP-SCO with heavy-tailed data in the high dimensional
space. In the first part we focus on the problem over some polytope constraint
(such as the $ell_1$-norm ball). We show that if the loss function is smooth
and its gradient has bounded second order moment, it is possible to get a (high
probability) error bound (excess population risk) of $tilde{O}(frac{log
d}{(nepsilon)^frac{1}{3}})$ in the $epsilon$-DP model, where $n$ is the
sample size and $d$ is the dimensionality of the underlying space. Next, for
LASSO, if the data distribution that has bounded fourth-order moments, we
improve the bound to $tilde{O}(frac{log d}{(nepsilon)^frac{2}{5}})$ in the
$(epsilon, delta)$-DP model. In the second part of the paper, we study sparse
learning with heavy-tailed data. We first revisit the sparse linear model and
propose a truncated DP-IHT method whose output could achieve an error of
$tilde{O}(frac{s^{*2}log d}{nepsilon})$, where $s^*$ is the sparsity of the
underlying parameter. Then we study a more general problem over the sparsity
({em i.e.,} $ell_0$-norm) constraint, and show that it is possible to achieve
an error of $tilde{O}(frac{s^{*frac{3}{2}}log d}{nepsilon})$, which is
also near optimal up to a factor of $tilde{O}{(sqrt{s^*})}$, if the loss
function is smooth and strongly convex.