Distribution Testing with a Confused Collector

We are interested in testing properties of distributions with systematically mislabeled samples. Our goal is to make decisions about unknown probability distributions, using a sample that has been collected by a confused collector, such as a machine-learning classifier that has not learned to distinguish all elements of the domain. The confused collector holds an unknown clustering of the domain and an input distribution $\mu$, and provides two oracles: a sample oracle which produces a sample from $\mu$ that has been labeled according to the clustering; and a label-query oracle which returns the label of a query point $x$ according to the clustering. Our first set of results shows that identity, uniformity, and equivalence of distributions can be tested efficiently, under the earth-mover distance, with remarkably weak conditions on the confused collector, even when the unknown clustering is adversarial. This requires defining a variant of the distribution testing task (inspired by the recent testable learning framework of Rubinfeld & Vasilyan), where the algorithm should test a joint property of the distribution and its clustering. As an example, we get efficient testers when the distribution tester is allowed to reject if it detects that the confused collector clustering is "far" from being a decision tree. The second set of results shows that we can sometimes do significantly better when the clustering is random instead of adversarial. For certain one-dimensional random clusterings, we show that uniformity can be tested under the TV distance using $\widetilde O\left(\frac{\sqrt n}{\rho^{3/2} \epsilon^2}\right)$ samples and zero queries, where $\rho \in (0,1]$ controls the "resolution" of the clustering. We improve this to $O\left(\frac{\sqrt n}{\rho \epsilon^2}\right)$ when queries are allowed.