Communication Lower Bounds for Statistical Estimation Problems via a Distributed Data Processing Inequality
This work addresses communication efficiency in distributed machine learning for high-dimensional data, offering foundational insights that could impact all of ML/AI, though it is incremental in extending previous lower bounds.
The paper tackles the tradeoff between statistical error and communication cost in distributed statistical estimation problems, such as sparse Gaussian mean estimation and sparse linear regression, by providing tight lower bounds on communication bits required to achieve minimax error, improving upon prior work by allowing multi-round iterative communication.
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 \textit{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 [Sha14,SD'14] 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 \textit{distributed data processing inequality}, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.