This theoretical note explores statistical versus computational trade-off to address a basic question in the application of divide-and-conquer method: what is the minimal computational cost in obtaining statistical optimality? In smoothing spline setup, we observe a phase transition phenomenon for the number of deployed machines that ends up being a simple proxy for computing cost. Specifically, a sharp upper bound for the number of machines is established: when the number is below this bound, statistical optimality (in terms of nonparametric estimation or testing) is achievable; otherwise, statistical optimality becomes impossible. These sharp bounds capture intrinsic computational limits of divide-and-conquer method, which turn out to be fully determined by the smoothness of the regression function. As a side remark, we argue that sample spitting may be viewed as a new form of regularization, playing a similar role as smoothing parameter.
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