  Finding efficient, easily implementable differentially private (DP)
algorithms that offer strong excess risk bounds is an important problem in
modern machine learning. To date, most work has focused on private empirical
risk minimization (ERM) or private population loss minimization. However, there
are often other objectives--such as fairness, adversarial robustness, or
sensitivity to outliers--besides average performance that are not captured in
the classical ERM setup. To this end, we study a completely general family of
convex, Lipschitz loss functions and establish the first known DP excess risk
and runtime bounds for optimizing this broad class. We provide similar bounds
under additional assumptions of smoothness and/or strong convexity. We also
address private stochastic convex optimization (SCO). While $(\epsilon,
\delta)$-DP ($\delta > 0$) has been the focus of much recent work in private
SCO, proving tight population loss bounds and runtime bounds for $(\epsilon,
0)$-DP remains a challenging open problem. We provide the tightest known
$(\epsilon, 0)$-DP population loss bounds and fastest runtimes under the
presence of (or lack of) smoothness and strong convexity. Our methods extend to
the $\delta > 0$ setting, where we offer the unique benefit of ensuring
differential privacy for arbitrary $\epsilon > 0$ by incorporating a new form
of Gaussian noise. Finally, we apply our theory to two learning frameworks:
tilted ERM and adversarial learning. In particular, our theory quantifies
tradeoffs between adversarial robustness, privacy, and runtime. Our results are
achieved using perhaps the simplest DP algorithm: output perturbation. Although
this method is not novel conceptually, our novel implementation scheme and
analysis show that the power of this method to achieve strong privacy, utility,
and runtime guarantees has not been fully appreciated in prior works.
