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- W3204583247 abstract "We consider the problem of estimating a $d$-dimensional discrete distribution from its samples observed under a $b$-bit communication constraint. In contrast to most previous results that largely focus on the global minimax error, we study the local behavior of the estimation error and provide emph{pointwise} bounds that depend on the target distribution $p$. In particular, we show that the $ell_2$ error decays with $Oleft(frac{lVert prVert_{1/2}}{n2^b}vee frac{1}{n}right)$ (In this paper, we use $avee b$ and $a wedge b$ to denote $max(a, b)$ and $min(a,b)$ respectively.) when $n$ is sufficiently large, hence it is governed by the emph{half-norm} of $p$ instead of the ambient dimension $d$. For the achievability result, we propose a two-round sequentially interactive estimation scheme that achieves this error rate uniformly over all $p$. Our scheme is based on a novel local refinement idea, where we first use a standard global minimax scheme to localize $p$ and then use the remaining samples to locally refine our estimate. We also develop a new local minimax lower bound with (almost) matching $ell_2$ error, showing that any interactive scheme must admit a $Omegaleft( frac{lVert p rVert_{{(1+delta)}/{2}}}{n2^b}right)$ $ell_2$ error for any $delta > 0$. The lower bound is derived by first finding the best parametric sub-model containing $p$, and then upper bounding the quantized Fisher information under this model. Our upper and lower bounds together indicate that the $mathcal{H}_{1/2}(p) = log(lVert p rVert_{{1}/{2}})$ bits of communication is both sufficient and necessary to achieve the optimal (centralized) performance, where $mathcal{H}_{{1}/{2}}(p)$ is the R'enyi entropy of order $2$. Therefore, under the $ell_2$ loss, the correct measure of the local communication complexity at $p$ is its R'enyi entropy." @default.
- W3204583247 created "2021-10-11" @default.
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- W3204583247 date "2021-10-07" @default.
- W3204583247 modified "2023-09-27" @default.
- W3204583247 title "Pointwise Bounds for Distribution Estimation under Communication Constraints" @default.
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