  We revisit the well-studied problem of differentially private empirical risk
minimization (ERM). We show that for unconstrained convex generalized linear
models (GLMs), one can obtain an excess empirical risk of $\tilde
O\left(\sqrt{{\texttt{rank}}}/\epsilon n\right)$, where ${\texttt{rank}}$ is
the rank of the feature matrix in the GLM problem, $n$ is the number of data
samples, and $\epsilon$ is the privacy parameter. This bound is attained via
differentially private gradient descent (DP-GD). Furthermore, via the first
lower bound for unconstrained private ERM, we show that our upper bound is
tight. In sharp contrast to the constrained ERM setting, there is no dependence
on the dimensionality of the ambient model space ($p$). (Notice that
${\texttt{rank}}\leq \min\{n, p\}$.) Besides, we obtain an analogous excess
population risk bound which depends on ${\texttt{rank}}$ instead of $p$.
  For the smooth non-convex GLM setting (i.e., where the objective function is
non-convex but preserves the GLM structure), we further show that DP-GD attains
a dimension-independent convergence of $\tilde
O\left(\sqrt{{\texttt{rank}}}/\epsilon n\right)$ to a
first-order-stationary-point of the underlying objective.
  Finally, we show that for convex GLMs, a variant of DP-GD commonly used in
practice (which involves clipping the individual gradients) also exhibits the
same dimension-independent convergence to the minimum of a well-defined
objective. To that end, we provide a structural lemma that characterizes the
effect of clipping on the optimization profile of DP-GD.
