On Communication Complexity of Classification Problems
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Proceedings of the ThirtySecond Conference on Learning Theory, PMLR 99:19031943, 2019.
Abstract
This work studies distributed learning in the spirit of Yao’s model of communication complexity: consider a twoparty 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 halfspaces in $\R^d$, 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 propercase as well as in the impropercase. These findings imply unconditional separations in this context between various learning tasks, e.g. realizable versus agnostic learning, proper versus improper learning, etcetera. %They also imply lower bounds that match the performance %of algorithm from previous works. 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 prespecified class. From a technical perspective, the protocols we devise (i.e. the upper bounds) are based on ideas from machine learning and the impossibility results (i.e. the lower bounds) are based on ideas from communication complexity.
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