  We consider parameter estimation in distributed networks, where each sensor
in the network observes an independent sample from an underlying distribution
and has $k$ bits to communicate its sample to a centralized processor which
computes an estimate of a desired parameter. We develop lower bounds for the
minimax risk of estimating the underlying parameter for a large class of losses
and distributions. Our results show that under mild regularity conditions, the
communication constraint reduces the effective sample size by a factor of $d$
when $k$ is small, where $d$ is the dimension of the estimated parameter.
Furthermore, this penalty reduces at most exponentially with increasing $k$,
which is the case for some models, e.g., estimating high-dimensional
distributions. For other models however, we show that the sample size reduction
is re-mediated only linearly with increasing $k$, e.g. when some sub-Gaussian
structure is available. We apply our results to the distributed setting with
product Bernoulli model, multinomial model, Gaussian location models, and
logistic regression which recover or strengthen existing results.
  Our approach significantly deviates from existing approaches for developing
information-theoretic lower bounds for communication-efficient estimation. We
circumvent the need for strong data processing inequalities used in prior work
and develop a geometric approach which builds on a new representation of the
communication constraint. This approach allows us to strengthen and generalize
existing results with simpler and more transparent proofs.
