The CEO Problem with $r$th Power of Difference and Logarithmic Distortions

The CEO problem has received much attention since first introduced by Berger et al., but there are limited results on non-Gaussian models with non-quadratic distortion measures. In this work, we extend the quadratic Gaussian CEO problem to two non-Gaussian settings with general $r$th power of difference distortion. Assuming an identical observation channel across agents, we study the asymptotics of distortion decay as the number of agents and sum-rate, $R_{sum}$, grow without bound, while individual rates vanish. The first setting is a regular source-observation model with $r$th power of difference distortion, which subsumes the quadratic Gaussian CEO problem, and we establish that the distortion decays at $\mathcal{O}(R_{sum}^{-r/2})$ when $r \ge 2$. We use sample median estimation after the Berger-Tung scheme for achievability. The other setting is a \emph{non-regular} source-observation model, including uniform additive noise models, with $r$th power of difference distortion for which estimation-theoretic regularity conditions do not hold. The distortion decay $\mathcal{O}(R_{sum}^{-r})$ when $r \ge 1$ is obtained for the non-regular model by midrange estimator following the Berger-Tung scheme. We also provide converses based on the Shannon lower bound for the regular model and the Chazan-Zakai-Ziv bound for the non-regular model, respectively. Lastly, we provide a sufficient condition for the regular model, under which quadratic and logarithmic distortions are asymptotically equivalent by an entropy power relationship as the number of agents grows. This proof relies on the Bernstein-von Mises theorem.