Distributed function estimation: adaptation using minimal communication

TL;DR

This paper explores the limits of adaptive function estimation in distributed systems with minimal communication, revealing that optimal rates are achievable for some risks but not others, depending on system parameters.

Contribution

It demonstrates the feasibility of adaptive estimation under minimal communication for the $L_2$-risk in certain regimes, and shows impossibility for the $L_ abla$-risk.

Findings

Optimal rates are achievable for $L_2$-risk under certain conditions.

Adaptive estimation for $L_ abla$-risk is impossible with minimal communication.

The feasibility depends on the relation between number of servers and total sample size.

Abstract

We investigate whether in a distributed setting, adaptive estimation of a smooth function at the optimal rate is possible under minimal communication. It turns out that the answer depends on the risk considered and on the number of servers over which the procedure is distributed. We show that for the $L_\infty$-risk, adaptively obtaining optimal rates under minimal communication is not possible. For the $L_2$-risk, it is possible over a range of regularities that depends on the relation between the number of local servers and the total sample size.