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We study hypothesis testing under communication constraints, where eachsample is quantized before being revealed to a statistician. Withoutcommunication constraints, it is well known that the sample complexity ofsimple binary hypothesis testing is characterized by the Hellinger distancebetween the distributions. We show that the sample complexity of simple binaryhypothesis testing under communication constraints is at most a logarithmicfactor larger than in the unconstrained setting and this bound is tight. Wedevelop a polynomial-time algorithm that achieves the aforementioned samplecomplexity. Our framework extends to robust hypothesis testing, where thedistributions are corrupted in the total variation distance. Our proofs rely ona new reverse data processing inequality and a reverse Markov inequality, whichmay be of independent interest. For simple $M$-ary hypothesis testing, thesample complexity in the absence of communication constraints has a logarithmicdependence on $M$. We show that communication constraints can cause anexponential blow-up leading to $\Omega(M)$ sample complexity even for adaptivealgorithms.

Communication-constrained hypothesis testing: Optimality, robustness, and reverse data processing inequalities