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We consider the problem of learning high-dimensional, nonparametric andstructured (e.g. Gaussian) distributions in distributed networks, where eachnode in the network observes an independent sample from the underlyingdistribution and can use $k$ bits to communicate its sample to a centralprocessor. We consider three different models for communication. Under theindependent model, each node communicates its sample to a central processor byindependently encoding it into $k$ bits. Under the more general sequential orblackboard communication models, nodes can share information interactively buteach node is restricted to write at most $k$ bits on the final transcript. Wecharacterize the impact of the communication constraint $k$ on the minimax riskof estimating the underlying distribution under $\ell^2$ loss. We developminimax lower bounds that apply in a unified way to many common statisticalmodels and reveal that the impact of the communication constraint can bequalitatively different depending on the tail behavior of the score functionassociated with each model. A key ingredient in our proofs is a geometriccharacterization of Fisher information from quantized samples.

Lower Bounds for Learning Distributions under Communication Constraints via Fisher Information