Information-theoretic lower bounds for distributed statistical estimation with communication constraints. Zhang, Y., Duchi, J., Jordan, M. I., & Wainwright, M. J. In Advances in Neural Information Processing Systems 26, NeurIPS'13, pages 2328–2336, 2013.
[ZDJW13] and [DJWZ14] Uses strong data processing inequalities (SDPI) to obtain lower bounds for simultaneous message passing (SMP) protocols under communication constraints under $\ell_2$ loss for various estimation problems, including Gaussian and Bernoulli mean estimation. Also includes some results for interactive protocols.

Information-theoretic lower bounds for distributed statistical estimation with communication constraints [link]Paper  bibtex   
@inproceedings{ZDJW13,
  title={Information-theoretic lower bounds for distributed statistical estimation with communication constraints},
  author={Zhang, Yuchen and Duchi, John and Jordan, Michael I. and Wainwright, Martin J.},
  booktitle= {Advances in Neural Information Processing Systems 26, {NeurIPS'13}},
  pages={2328--2336},
  year={2013},
  url = {https://arxiv.org/abs/1405.0782},
  bibbase_note = {<div class="well well-small bibbase"><span class="bluecite">[ZDJW13]</span> and <span class="bluecite">[DJWZ14]</span> Uses strong data processing inequalities (SDPI) to obtain lower bounds for simultaneous message passing (SMP) protocols  under communication constraints under $\ell_2$ loss for various estimation problems, including Gaussian and Bernoulli mean estimation. Also includes some results for interactive protocols.</div>}
}

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