Geometric Lower Bounds for Distributed Parameter Estimation Under Communication Constraints

Over-the-Air Statistical Estimation

“…In particular, recent work in machine learning [1]- [6] has studied the impact of communication constraints on distributed parameter estimation. These works abstract out the physical layer, simply assuming a constraint on the number of bits available to represent each sample.…”

“…We can also derive a result for product Bernoulli models where elements of are close to 1 2 , i.e., where the samples are dense. Corollary 4.…”

“…. , ∼ =1 Bernoulli( ), with Θ = [ 1 2 − , 1 2 + ] , ∈ (0, 1 2 ), with channel uses. The worst-case risk under squared error loss of any estimation scheme (f, ˆ ) must satisfy…”

“…One simple and intuitive way to study the impact of bandwidth-limitations on estimation performance is to model them as imposing constraints on the number of bits available to encode each observed sample. A number of works in the machine learning literature have taken this tack [1]- [6], providing both achievable schemes and informationtheoretic lower bounds under bit constraints, and characterizing the dependence of the estimation error on the number of bits available to represent each sample. Since these analyses assume the encoded bits to be received without error, they can be interpreted as assuming that a reliable scheme is used for transmission over the underlying noisy channels.…”

“…First, while the lower bounds we presented are general, our achievability results pertain specifically to the two estimation models we studied. With the additive nature of the Gaussian MAC, one would imagine that this extends to other mean estimation problems, but it raises the question of whether other estimation problems, such as distribution or empirical frequency estimation [6], [42], [43], can also harness this or other channels to achieve similarly startling gains over digital schemes.…”