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arXiv 1802.08417cs.DCcs.ITmath.ITstat.ME
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通信约束下分布参数估计的几何下界

Geometric Lower Bounds for Distributed Parameter Estimation under Communication Constraints

Yanjun Han, Ayfer Özgür, Tsachy Weissman

摘要:

我们认为在分布式网络中的参数估计,其中网络中的每个传感器观察到一个独立的样本从一个底层的分布,并有$k$位通信的样本到一个集中的处理器计算所需的参数估计。我们开发了一个大类的损失和分布的极小极大风险估计的基本参数的下界。我们的研究结果表明,在温和的规则性条件下,通信约束减少了有效样本量的一个因素$d$时,$k$是小的,其中$d$是估计参数的维数。此外,该惩罚最多随着$k$的增加而指数地减小,这是某些模型的情况,例如,估计高维分布。然而,对于其他模型,我们表明,重新介导的样本大小的减少,只有线性增加$k$,例如,当一些亚高斯结构是可用的。我们将我们的结果应用到分布设置与产品伯努利模型,多项式模型,高斯位置模型,和逻辑回归恢复或加强现有的结果。我们的方法显着偏离现有的方法开发信息理论的通信效率估计的下限。我们规避了需要强有力的数据处理不平等在以前的工作中使用,并开发了一种几何方法,它建立在一个新的表示的通信约束。这种方法使我们能够加强和推广现有的结果更简单,更透明的证明。

英文摘要:

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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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