On Communication Complexity of Classification Problems

TL;DR

This paper investigates the communication complexity in distributed learning scenarios, providing characterizations of learnable classes and demonstrating fundamental separations between different learning models using a novel framework.

Contribution

It introduces a unified framework for analyzing communication complexity in distributed learning, with combinatorial characterizations and new separations between learning paradigms.

Findings

Characterization of classes learnable with efficient communication

Unconditional separations between realizable and agnostic learning

Unconditional separations between proper and improper learning

Abstract

This work studies distributed learning in the spirit of Yao's model of communication complexity: consider a two-party setting, where each of the players gets a list of labelled examples and they communicate in order to jointly perform some learning task. To naturally fit into the framework of learning theory, the players can send each other examples (as well as bits) where each example/bit costs one unit of communication. This enables a uniform treatment of infinite classes such as half-spaces in $\mathbb{R}^d$, which are ubiquitous in machine learning. We study several fundamental questions in this model. For example, we provide combinatorial characterizations of the classes that can be learned with efficient communication in the proper-case as well as in the improper-case. These findings imply unconditional separations between various learning contexts, e.g.\ realizable versus agnostic learning, proper versus improper learning, etc. The derivation of these results hinges on a type of decision problems we term "{\it realizability problems}" where the goal is deciding whether a distributed input sample is consistent with an hypothesis from a pre-specified class. From a technical perspective, the protocols we use are based on ideas from machine learning theory and the impossibility results are based on ideas from communication complexity theory.