High-Dimensional Multi-Task Averaging and Application to Kernel Mean Embedding
This work addresses the challenge of improving estimation accuracy in high-dimensional multi-task settings, with applications in kernel mean embeddings, though it appears incremental as it builds on existing multi-task averaging methods.
The paper tackles the multi-task averaging problem of jointly estimating means of multiple distributions from separate datasets by proposing an estimator that shrinks naive empirical means towards local averages of similar tasks, demonstrating a significant reduction in mean squared error, especially in high dimensions.
We propose an improved estimator for the multi-task averaging problem, whose goal is the joint estimation of the means of multiple distributions using separate, independent data sets. The naive approach is to take the empirical mean of each data set individually, whereas the proposed method exploits similarities between tasks, without any related information being known in advance. First, for each data set, similar or neighboring means are determined from the data by multiple testing. Then each naive estimator is shrunk towards the local average of its neighbors. We prove theoretically that this approach provides a reduction in mean squared error. This improvement can be significant when the dimension of the input space is large, demonstrating a "blessing of dimensionality" phenomenon. An application of this approach is the estimation of multiple kernel mean embeddings, which plays an important role in many modern applications. The theoretical results are verified on artificial and real world data.