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Change point estimation is often formulated as a search for the maximum of again function describing improved fits when segmenting the data. Searchingthrough all candidates requires $O(n)$ evaluations of the gain function for aninterval with $n$ observations. If each evaluation is computationally demanding(e.g. in high-dimensional models), this can become infeasible. Instead, wepropose optimistic search methods with $O(\log n)$ evaluations exploitingspecific structure of the gain function.  Towards solid understanding of our strategy, we investigate in detail the$p$-dimensional Gaussian changing means setup, including high-dimensionalscenarios. For some of our proposals, we prove asymptotic minimax optimalityfor detecting change points and derive their asymptotic localization rate.These rates (up to a possible log factor) are optimal for the univariate andmultivariate scenarios, and are by far the fastest in the literature under theweakest possible detection condition on the signal-to-noise ratio in thehigh-dimensional scenario. Computationally, our proposed methodology has theworst case complexity of $O(np)$, which can be improved to be sublinear in $n$if some a-priori knowledge on the length of the shortest segment is available.  Our search strategies generalize far beyond the theoretically analyzed setup.We illustrate, as an example, massive computational speedup in change pointdetection for high-dimensional Gaussian graphical models.

Optimistic search: Change point estimation for large-scale data via adaptive logarithmic queries