Sparse Normal Means Estimation with Sublinear Communication

• URL: http://arxiv.org/abs/2102.03060v1
• Date: Fri, 5 Feb 2021 08:52:25 GMT
• Title: Sparse Normal Means Estimation with Sublinear Communication
• Authors: Chen Amiraz, Robert Krauthgamer, Boaz Nadler
• Abstract summary: We consider the problem of normal means estimation in a distributed setting with communication constraints. We show that once the signal-to-noise ratio (SNR) is slightly higher, the support of $mu$ can be correctly recovered with much less communication.
• Score: 13.257075075199781
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of sparse normal means estimation in a distributed setting with communication constraints. We assume there are $M$ machines, each holding a $d$-dimensional observation of a $K$-sparse vector $\mu$ corrupted by additive Gaussian noise. A central fusion machine is connected to the $M$ machines in a star topology, and its goal is to estimate the vector $\mu$ with a low communication budget. Previous works have shown that to achieve the centralized minimax rate for the $\ell_2$ risk, the total communication must be high - at least linear in the dimension $d$. This phenomenon occurs, however, at very weak signals. We show that once the signal-to-noise ratio (SNR) is slightly higher, the support of $\mu$ can be correctly recovered with much less communication. Specifically, we present two algorithms for the distributed sparse normal means problem, and prove that above a certain SNR threshold, with high probability, they recover the correct support with total communication that is sublinear in the dimension $d$. Furthermore, the communication decreases exponentially as a function of signal strength. If in addition $KM\ll d$, then with an additional round of sublinear communication, our algorithms achieve the centralized rate for the $\ell_2$ risk. Finally, we present simulations that illustrate the performance of our algorithms in different parameter regimes.

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