Locally private online change point detection

Abstract: We study online change point detection problems under the constraint of local differential privacy (LDP) where, in particular, the statistician does not have access to the raw data. As a concrete problem, we study a multivariate nonparametric regression problem. At each time point $t$, the raw data are assumed to be of the form $(X_t, Y_t)$, where $X_t$ is a $d$-dimensional feature vector and $Y_t$ is a response variable. Our primary aim is to detect changes in the regression function $m_t(x)=\mathbb{E}(Y_t |X_t=x)$ as soon as the change occurs. We provide algorithms which respect the LDP constraint, which control the false alarm probability, and which detect changes with a minimal (minimax rate-optimal) delay. To quantify the cost of privacy, we also present the optimal rate in the benchmark, non-private setting. These non-private results are also new to the literature and thus are interesting \emph{per se}. In addition, we study the univariate mean online change point detection problem, under privacy constraints. This serves as the blueprint of studying more complicated private change point detection problems.