  In this paper, we consider the problem of designing Differentially Private
(DP) algorithms for Stochastic Convex Optimization (SCO) on heavy-tailed data.
The irregularity of such data violates some key assumptions used in almost all
existing DP-SCO and DP-ERM methods, resulting in failure to provide the DP
guarantees. To better understand this type of challenges, we provide in this
paper a comprehensive study of DP-SCO under various settings. First, we
consider the case where the loss function is strongly convex and smooth. For
this case, we propose a method based on the sample-and-aggregate framework,
which has an excess population risk of $\tilde{O}(\frac{d^3}{n\epsilon^4})$
(after omitting other factors), where $n$ is the sample size and $d$ is the
dimensionality of the data. Then, we show that with some additional assumptions
on the loss functions, it is possible to reduce the \textit{expected} excess
population risk to $\tilde{O}(\frac{ d^2}{ n\epsilon^2 })$. To lift these
additional conditions, we also provide a gradient smoothing and trimming based
scheme to achieve excess population risks of $\tilde{O}(\frac{
d^2}{n\epsilon^2})$ and
$\tilde{O}(\frac{d^\frac{2}{3}}{(n\epsilon^2)^\frac{1}{3}})$ for strongly
convex and general convex loss functions, respectively, \textit{with high
probability}. Experiments suggest that our algorithms can effectively deal with
the challenges caused by data irregularity.
