Exploring the Gap between Tolerant and Non-tolerant Distribution Testing

The framework of distribution testing is currently ubiquitous in the field of property testing. In this model, the input is a probability distribution accessible via independently drawn samples from an oracle. The testing task is to distinguish a distribution that satisfies some property from a distribution that is far from satisfying it in the $\ell_1$ distance. The task of tolerant testing imposes a further restriction, that distributions close to satisfying the property are also accepted. This work focuses on the connection of the sample complexities of non-tolerant ("traditional") testing of distributions and tolerant testing thereof. When limiting our scope to label-invariant (symmetric) properties of distribution, we prove that the gap is at most quadratic. Conversely, the property of being the uniform distribution is indeed known to have an almost-quadratic gap. When moving to general, not necessarily label-invariant properties, the situation is more complicated, and we show some partial results. We show that if a property requires the distributions to be non-concentrated, then it cannot be non-tolerantly tested with $o(\sqrt{n})$ many samples, where $n$ denotes the universe size. Clearly, this implies at most a quadratic gap, because a distribution can be learned (and hence tolerantly tested against any property) using $\mathcal{O}(n)$ many samples. Being non-concentrated is a strong requirement on the property, as we also prove a close to linear lower bound against their tolerant tests. To provide evidence for other general cases (where the properties are not necessarily label-invariant), we show that if an input distribution is very concentrated, in the sense that it is mostly supported on a subset of size $s$ of the universe, then it can be learned using only $\mathcal{O}(s)$ many samples. The learning procedure adapts to the input, and works without knowing $s$ in advance.