Communication Complexity of Two-party Nonparametric Global Density Estimation

Liu, Jingbo

doi:10.1109/ciss53076.2022.9751150

Cited by 2 publications

(2 citation statements)

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“…To achieve the scaling in (28), r can grow as slowly as the super-logarithm (i.e., inverse tetration) of k; for the precise relation between r and k, see Section V-C.…”

Section: Problem Setup and Main Resultsmentioning

confidence: 99%

“…Let us emphasize that this paper concerns density estimation at a given point, rather than estimating the global density function. For the latter problem, it is optimal for Alice to just quantize the samples and send it to Bob, which we show in the companion paper [28] . The mean square error (in 2 norm) of estimating global density function scales differently than the case of point-wise density estimation since the messages cannot be tailored to the given point.…”

Section: Introductionmentioning

confidence: 97%

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A Few Interactions Improve Distributed Nonparametric Estimation, Optimally

Liu

2023

IEEE Trans. Inform. Theory

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Consider the problem of nonparametric estimation of an unknown β-Hölder smooth density pXY at a given point, where X and Y are both d dimensional. An infinite sequence of i.i.d. samples (Xi, Yi) are generated according to this distribution, and two terminals observe (Xi) and (Yi), respectively. They are allowed to exchange k bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean square risk is orderd+2β for one-way protocols and kd+2β for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of k. The proof of the upper bound is based on a novel multiround scheme for estimating the joint distribution of a pair of biased Bernoulli variables, and the lower bound is built on a sharp estimate of a symmetric strong data processing constant for biased Bernoulli variables.

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“…To achieve the scaling in (28), r can grow as slowly as the super-logarithm (i.e., inverse tetration) of k; for the precise relation between r and k, see Section V-C.…”

Section: Problem Setup and Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%