## Alice and Bob Show Distribution Testing Lower Bounds (They don't talk to each other anymore)

by Eric Blais, Clement Canonne, and Tom Gur

Let me highlight two contributions of this paper, although the paper's abstract kind of does so. The first contribution is the introduction of a technique for proving lower bounds on the sample complexity of distribution testing. The second is in providing an alternative characterization of the sample complexity of testing equality of a given distribution to a fixed distribution, where the characterization is in terms of a natural measure'' of the fixed distribution (i.e., the size of its effective'' support).
Our main result is concerned with testing identity to a specific distribution $p$, given as a parameter. In a recent and influential work, Valiant and Valiant [VV14] showed that the sample complexity of the aforementioned problem is closely related to the $\ell_{2/3}$-quasinorm of $p$. We obtain alternative bounds on the complexity of this problem in terms of an arguably more intuitive measure and using simpler proofs. More specifically, we prove that the sample complexity is essentially determined by a fundamental operator in the theory of interpolation of Banach spaces, known as Peetre's $K$-functional. We show that this quantity is closely related to the size of the effective support of $p$ (loosely speaking, the number of supported elements that constitute the vast majority of the mass of $p)$. This result, in turn, stems from an unexpected connection to functional analysis and refined concentration of measure inequalities, which arise naturally in our reduction.