The Price of Tolerance in Distribution Testing
Abstract
We revisit the problem of tolerant distribution testing. That is, given samples from an unknown distribution $p$ over $\{1, \dots, n\}$, is it $\varepsilon_1$close to or $\varepsilon_2$far from a reference distribution $q$ (in total variation distance)? Despite significant interest over the past decade, this problem is well understood only in the extreme cases. In the noiseless setting (i.e., $\varepsilon_1 = 0$) the sample complexity is $\Theta(\sqrt{n})$, strongly sublinear in the domain size. At the other end of the spectrum, when $\varepsilon_1 = \varepsilon_2/2$, the sample complexity jumps to the barely sublinear $\Theta(n/\log n)$. However, very little is known about the intermediate regime. We fully characterize the price of tolerance in distribution testing as a function of $n$, $\varepsilon_1$, $\varepsilon_2$, up to a single $\log n$ factor. Specifically, we show the sample complexity to be \[\tilde \Theta\left(\frac{\sqrt{n}}{\varepsilon_2^{2}} + \frac{n}{\log n} \cdot \max \left\{\frac{\varepsilon_1}{\varepsilon_2^2},\left(\frac{\varepsilon_1}{\varepsilon_2^2}\right)^{\!\!2}\right\}\right),\] providing a smooth tradeoff between the two previously known cases. We also provide a similar characterization for the problem of tolerant equivalence testing, where both $p$ and $q$ are unknown. Surprisingly, in both cases, the main quantity dictating the sample complexity is the ratio $\varepsilon_1/\varepsilon_2^2$, and not the more intuitive $\varepsilon_1/\varepsilon_2$. Of particular technical interest is our lower bound framework, which involves novel approximationtheoretic tools required to handle the asymmetry between $\varepsilon_1$ and $\varepsilon_2$, a challenge absent from previous works.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.13414
 Bibcode:
 2021arXiv210613414C
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Information Theory;
 Mathematics  Probability;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning
 EPrint:
 Added a result on instanceoptimal testing, and further discussion in the introduction