Distributed Gaussian Mean Testing under Communication Constraints: messages, samples, and coins
It provides a comprehensive characterization of the trade-offs between communication, samples, and shared randomness for a fundamental statistical testing problem, with implications for distributed inference.
The paper generalizes distributed Gaussian mean testing to settings with limited shared randomness, heterogeneous sample sizes, and heterogeneous communication budgets, providing tight lower bounds and matching protocols.
We revisit the problem of Gaussian mean testing in a distributed, communication constrained setting, where each of $n$ users independently observes samples from an unknown $d$-dimensional spherical Gaussian distribution $\mathcal{G}(μ,\mathbb{I}_d)$, and can communicate up to $\ell$ bits to a central referee. The referee's goal is then to distinguish between cases (i) $\|μ\|_2 = 0$ versus (ii) $\|μ\|_2\ge \varepsilon$. This problem has been considered in the private- and public-coin settings, when each user holds exactly one sample, or more generally when each holds exactly $m$ samples. In this work, we significantly generalize the question in three directions: when the users only share a small number $s$ of random bits, when each user holds a different number of samples $m_k$, and when each user can send a different number of bits $\ell_k$ to the referee.