Paper Info
Title
Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality.
Abstract
We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the m machines receives n data points from a d-dimensional Gaussian distribution with unknown mean θ which is promised to be k-sparse. The machines communicate by message passing and aim to estimate the mean θ. We provide a tight (up to logarithmic factors) tradeoff between the estimation error and the number of bits communicated between the machines. This directly leads to a lower bound for the distributed sparse linear regression problem: to achieve the statistical minimax error, the total communication is at least Ω(min{n,d}m), where n is the number of observations that each machine receives and d is the ambient dimension. These lower results improve upon Shamir (NIPS'14) and Steinhardt-Duchi (COLT'15) by allowing multi-round iterative communication model. We also give the first optimal simultaneous protocol in the dense case for mean estimation. As our main technique, we prove a distributed data processing inequality, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.
Year
DOI
Venue
2016
10.1145/2897518.2897582
STOC
Keywords
DocType
Volume
Communication complexity,Information complexity,statistical estimation
Conference
abs/1506.07216
ISSN
Citations 
PageRank 
0737-8017
19
0.77
References 
Authors
13
5
Name
Order
Citations
PageRank
Mark Braverman181061.60
Ankit Garg212516.19
Tengyu Ma359641.23
Huy L. Nguyen437632.33
David P. Woodruff52156142.38