  In this paper, we study the information transmission problem under the
distributed learning framework, where each worker node is merely permitted to
transmit a $m$-dimensional statistic to improve learning results of the target
node. Specifically, we evaluate the corresponding expected population risk
(EPR) under the regime of large sample sizes. We prove that the performance can
be enhanced since the transmitted statistics contribute to estimating the
underlying distribution under the mean square error measured by the EPR norm
matrix. Accordingly, the transmitted statistics correspond to the eigenvectors
of this matrix, and the desired transmission allocates these eigenvectors among
the statistics such that the EPR is minimal. Moreover, we provide the
analytical solution of the desired statistics for single-node and two-node
transmission, where a geometrical interpretation is given to explain the
eigenvector selection. For the general case, an efficient algorithm that can
output the allocation solution is developed based on the node partitions.
