Univariate Mean Change Point Detection: Penalization, CUSUM and Optimality

The problem of univariate mean change point detection and localization based on a sequence of $n$ independent observations with piecewise constant means has been intensively studied for more than half century, and serves as a blueprint for change point problems in more complex settings. We provide a complete characterization of this classical problem in a general framework in which the upper bound on the noise variance $\sigma^2$, the minimal spacing $\Delta$ between two consecutive change points and the minimal magnitude of the changes $\kappa$, are allowed to vary with $n$. We first show that consistent localization of the change points when the signal-to-noise ratio $\frac{\kappa \sqrt{\Delta}}{\sigma}$ is uniformly bounded from above is impossible. In contrast, when $\frac{\kappa \sqrt{\Delta}}{\sigma}$ is diverging in $n$ at any arbitrary slow rate, we demonstrate that two computationally-efficient change point estimators, one based on the solution to an $\ell_0$-penalized least squares problem and the other on the popular WBS algorithm, are both consistent and achieve a localization rate of the order $\frac{\sigma^2}{\kappa^2} \log(n)$. We further show that such rate is minimax optimal, up to a $\log(n)$ term.