Statistically and Computationally Efficient Change Point Localization in Regression Settings
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Detecting when the underlying distribution changes from the observed time series is a fundamental problem arising in a broad spectrum of applications. Change point localization is particularly challenging when we only observe low-dimensional projections of high-dimensional random variables. Specifically, we assume we observe $\{ x_t, y_t\}_{t=1}^n$ where $ \{ x_t\}_{t=1}^n$ are $p$-dimensional covariates, $\{y_t\}_{t=1}^n $ are the univariate responses satisfying $E(y_t)=x_t^\top \beta_t^*\text{ for all }1\let\le n$ and that $\{\beta_t^*\}_{t=1}^n $ are the unobserved regression parameters that change over time in a piecewise constant manner. We first propose a novel algorithm called Binary Segmentation through Estimated CUSUM statistics (BSE), which computes the change points through direct estimates of the CUSUM statistics of $\{\beta_t^*\}_{t=1}^n $. We show that BSE can consistently estimate the unknown location of the change points, achieving error bounds of order $O (\log(p)/n) $. To the best of our knowledge, this is a significant improvement, as the state-of-the-art methods are only shown to achieve error bounds of order $O(\log(p)/\sqrt n)$ in the multiple change point setting. However, BSE can be computationally costly. To overcome this limitation, we introduce another algorithm called Binary Segmentation through Lasso Estimators (BSLE). We show that BSLE can consistently localize change points with a slightly worse localization error rate compared to BSE, but BSLE is much more computationally efficient. Finally, we leverage the insights gained from BSE to develop a novel "local screening" algorithm that can input a coarse estimate of change point locations together with the observed data and efficiently refine that estimate, allowing us to improve the practical performance of past estimators. We also justify our theoretical finding in simulated experiments.
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