On learning parametric distributions from quantized samples
This work addresses a fundamental problem in distributed learning for sensor networks, but it appears incremental as it extends existing lower bound techniques.
The paper tackles the problem of estimating parametric distributions from quantized samples in a network, establishing minimax lower bounds on estimation error for L_p-norms and Wasserstein loss.
We consider the problem of learning parametric distributions from their quantized samples in a network. Specifically, $n$ agents or sensors observe independent samples of an unknown parametric distribution; and each of them uses $k$ bits to describe its observed sample to a central processor whose goal is to estimate the unknown distribution. First, we establish a generalization of the well-known van Trees inequality to general $L_p$-norms, with $p > 1$, in terms of Generalized Fisher information. Then, we develop minimax lower bounds on the estimation error for two losses: general $L_p$-norms and the related Wasserstein loss from optimal transport.