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We propose an improved estimator for the multi-task averaging problem, whosegoal is the joint estimation of the means of multiple distributions usingseparate, independent data sets. The naive approach is to take the empiricalmean of each data set individually, whereas the proposed method exploitssimilarities between tasks, without any related information being known inadvance. First, for each data set, similar or neighboring means are determinedfrom the data by multiple testing. Then each naive estimator is shrunk towardsthe local average of its neighbors. We prove theoretically that this approachprovides a reduction in mean squared error. This improvement can be significantwhen the dimension of the input space is large, demonstrating a "blessing ofdimensionality" phenomenon. An application of this approach is the estimationof multiple kernel mean embeddings, which plays an important role in manymodern applications. The theoretical results are verified on artificial andreal world data.

High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding