  We consider the problem of estimating a $d$-dimensional $s$-sparse discrete
distribution from its samples observed under a $b$-bit communication
constraint. The best-known previous result on $\ell_2$ estimation error for
this problem is $O\left( \frac{s\log\left( {d}/{s}\right)}{n2^b}\right)$.
Surprisingly, we show that when sample size $n$ exceeds a minimum threshold
$n^*(s, d, b)$, we can achieve an $\ell_2$ estimation error of $O\left(
\frac{s}{n2^b}\right)$. This implies that when $n>n^*(s, d, b)$ the convergence
rate does not depend on the ambient dimension $d$ and is the same as knowing
the support of the distribution beforehand.
  We next ask the question: ``what is the minimum $n^*(s, d, b)$ that allows
dimension-free convergence?''. To upper bound $n^*(s, d, b)$, we develop novel
localization schemes to accurately and efficiently localize the unknown
support. For the non-interactive setting, we show that $n^*(s, d, b) = O\left(
\min \left( {d^2\log^2 d}/{2^b}, {s^4\log^2 d}/{2^b}\right) \right)$. Moreover,
we connect the problem with non-adaptive group testing and obtain a
polynomial-time estimation scheme when $n = \tilde{\Omega}\left({s^4\log^4
d}/{2^b}\right)$. This group testing based scheme is adaptive to the sparsity
parameter $s$, and hence can be applied without knowing it. For the interactive
setting, we propose a novel tree-based estimation scheme and show that the
minimum sample-size needed to achieve dimension-free convergence can be further
reduced to $n^*(s, d, b) = \tilde{O}\left( {s^2\log^2 d}/{2^b} \right)$.
