Differentially Private Empirical Risk Minimization: Efficient Algorithms and Tight Error Bounds

TL;DR

This paper develops new differentially private algorithms for convex empirical risk minimization, providing tight error bounds and matching lower bounds, advancing the understanding of privacy-preserving machine learning.

Contribution

It introduces novel algorithms and lower bounds for private ERM under Lipschitz and strong convexity assumptions, with optimal or near-optimal efficiency and error guarantees.

Findings

Algorithms match non-private oracle complexity in some cases

Lower bounds apply to simple, smooth function families

Previous methods do not achieve optimal error rates for certain problems

Abstract

In this paper, we initiate a systematic investigation of differentially private algorithms for convex empirical risk minimization. Various instantiations of this problem have been studied before. We provide new algorithms and matching lower bounds for private ERM assuming only that each data point's contribution to the loss function is Lipschitz bounded 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 non-private running time (as measured by oracle complexity). We give separate algorithms (and lower bounds) for $(\epsilon,0)$- and $(\epsilon,\delta)$-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.