The Price of Tolerance in Distribution Testing

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 approximation-theoretic tools required to handle the asymmetry between \(\varepsilon_1\) and \(\varepsilon_2\), a challenge absent from previous works.