Differentially Private Empirical Risk Minimization with Non-convex Loss Functions

We study the problem of Empirical Risk Minimization (ERM) with (smooth) non-convex loss functions under the differential-privacy (DP) model. Existing approaches for this problem mainly adopt gradient norms to measure the error, which in general cannot guarantee the quality of the solution. To address this issue, we first study the expected excess empirical (or population) risk, which was primarily used as the utility to measure the quality for convex loss functions. Specifically, we show that the excess empirical (or population) risk can be upper bounded by $\tilde{O}(\frac{d\log (1/\delta)}{\log n\epsilon^2})$ in the $(\epsilon, \delta)$-DP settings, where $n$ is the data size and $d$ is the dimensionality of the space. The $\frac{1}{\log n}$ term in the empirical risk bound can be further improved to $\frac{1}{n^{\Omega(1)}}$ (when $d$ is a constant) by a highly non-trivial analysis on the time-average error. To obtain more efficient solutions, we also consider the connection between achieving differential privacy and finding approximate local minimum. Particularly, we show that when the size $n$ is large enough, there are $(\epsilon, \delta)$-DP algorithms which can find an approximate local minimum of the empirical risk with high probability in both the constrained and non-constrained settings. These results indicate that one can escape saddle points privately.