Convex empirical risk minimization is a basic tool in machine learning and statistics. We provide new algorithms and matching lower bounds for differentially private convex empirical risk minimization assuming only that each data point's contribution to the loss function is Lipschitz and that the domain of optimization is bounded. We provide a separate set of algorithms and matching lower bounds for the setting in which the loss functions are known to also be strongly convex. Our algorithms run in polynomial time, and in some cases even match the optimal nonprivate running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for ( , 0)and ( , δ)-differential privacy; perhaps surprisingly, the techniques used for designing optimal algorithms in the two cases are completely different. Our lower bounds apply even to very simple, smooth function families, such as linear and quadratic functions. This implies that algorithms from previous work can be used to obtain optimal error rates, under the additional assumption that the contributions of each data point to the loss function is smooth. We show that simple approaches to smoothing arbitrary loss functions (in order to apply previous techniques) do not yield optimal error rates. In particular, optimal algorithms were not previously known for problems such as training support vector machines and the high-dimensional median.
Bassily et al. REF proposed new algorithms for the private convex ERM problem when the loss function is only Lipschitz and the domain of the optimization is bounded.
Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
2014 IEEE 55th Annual Symposium on Foundations of Computer Science
Computer Science Mathematics