  We explore the connection between dimensionality and communication cost in
distributed learning problems. Specifically we study the problem of estimating
the mean $\vec{\theta}$ of an unknown $d$ dimensional gaussian distribution in
the distributed setting. In this problem, the samples from the unknown
distribution are distributed among $m$ different machines. The goal is to
estimate the mean $\vec{\theta}$ at the optimal minimax rate while
communicating as few bits as possible. We show that in this setting, the
communication cost scales linearly in the number of dimensions i.e. one needs
to deal with different dimensions individually. Applying this result to
previous lower bounds for one dimension in the interactive setting
\cite{ZDJW13} and to our improved bounds for the simultaneous setting, we prove
new lower bounds of $\Omega(md/\log(m))$ and $\Omega(md)$ for the bits of
communication needed to achieve the minimax squared loss, in the interactive
and simultaneous settings respectively. To complement, we also demonstrate an
interactive protocol achieving the minimax squared loss with $O(md)$ bits of
communication, which improves upon the simple simultaneous protocol by a
logarithmic factor. Given the strong lower bounds in the general setting, we
initiate the study of the distributed parameter estimation problems with
structured parameters. Specifically, when the parameter is promised to be
$s$-sparse, we show a simple thresholding based protocol that achieves the same
squared loss while saving a $d/s$ factor of communication. We conjecture that
the tradeoff between communication and squared loss demonstrated by this
protocol is essentially optimal up to logarithmic factor.
