  Differentially private (DP) stochastic convex optimization (SCO) is a
fundamental problem, where the goal is to approximately minimize the population
risk with respect to a convex loss function, given a dataset of $n$ i.i.d.
samples from a distribution, while satisfying differential privacy with respect
to the dataset. Most of the existing works in the literature of private convex
optimization focus on the Euclidean (i.e., $\ell_2$) setting, where the loss is
assumed to be Lipschitz (and possibly smooth) w.r.t. the $\ell_2$ norm over a
constraint set with bounded $\ell_2$ diameter. Algorithms based on noisy
stochastic gradient descent (SGD) are known to attain the optimal excess risk
in this setting.
  In this work, we conduct a systematic study of DP-SCO for $\ell_p$-setups
under a standard smoothness assumption on the loss. For $1< p\leq 2$, under a
standard smoothness assumption, we give a new, linear-time DP-SCO algorithm
with optimal excess risk. Previously known constructions with optimal excess
risk for $1< p <2$ run in super-linear time in $n$. For $p=1$, we give an
algorithm with nearly optimal excess risk. Our result for the $\ell_1$-setup
also extends to general polyhedral norms and feasible sets. Moreover, we show
that the excess risk bounds resulting from our algorithms for $1\leq p \leq 2$
are attained with high probability. For $2 < p \leq \infty$, we show that
existing linear-time constructions for the Euclidean setup attain a nearly
optimal excess risk in the low-dimensional regime. As a consequence, we show
that such constructions attain a nearly optimal excess risk for $p=\infty$. Our
work draws upon concepts from the geometry of normed spaces, such as the
notions of regularity, uniform convexity, and uniform smoothness.
