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Finding efficient, easily implementable differentially private (DP)algorithms that offer strong excess risk bounds is an important problem inmodern machine learning. To date, most work has focused on private empiricalrisk minimization (ERM) or private population loss minimization. However, thereare often other objectives--such as fairness, adversarial robustness, orsensitivity to outliers--besides average performance that are not captured inthe classical ERM setup. To this end, we study a completely general family ofconvex, Lipschitz loss functions and establish the first known DP excess riskand runtime bounds for optimizing this broad class. We provide similar boundsunder additional assumptions of smoothness and/or strong convexity. We alsoaddress private stochastic convex optimization (SCO). While $(\epsilon,\delta)$-DP ($\delta > 0$) has been the focus of much recent work in privateSCO, 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 thepresence of (or lack of) smoothness and strong convexity. Our methods extend tothe $\delta > 0$ setting, where we offer the unique benefit of ensuringdifferential privacy for arbitrary $\epsilon > 0$ by incorporating a new formof Gaussian noise. Finally, we apply our theory to two learning frameworks:tilted ERM and adversarial learning. In particular, our theory quantifiestradeoffs between adversarial robustness, privacy, and runtime. Our results areachieved using perhaps the simplest DP algorithm: output perturbation. Althoughthis method is not novel conceptually, our novel implementation scheme andanalysis show that the power of this method to achieve strong privacy, utility,and runtime guarantees has not been fully appreciated in prior works.

Output Perturbation for Differentially Private Convex Optimization with  Improved Population Loss Bounds, Runtimes and Applications to Private  Adversarial Training