Testing Self-Reducible Samplers

Samplers are the backbone of the implementations of any randomized algorithm. Unfortunately, obtaining an efficient algorithm to test the correctness of samplers is very hard to find. Recently, in a series of works, testers like Barbarik, Teq, Flash for testing of some particular kinds of samplers, like CNF-samplers and Horn-samplers, were obtained. However, their techniques have a significant limitation because one can not expect to use their methods to test for other samplers, such as perfect matching samplers or samplers for sampling linear extensions in posets. In this paper, we present a new testing algorithm that works for such samplers and can estimate the distance of a new sampler from a known sampler (say, the uniform sampler). Testing the identity of distributions is the heart of testing the correctness of samplers. This paper's main technical contribution is developing a new distance estimation algorithm for distributions over high-dimensional cubes using the recently proposed subcube conditioning sampling model. Given subcube conditioning access to an unknown distribution P, and a known distribution Q defined over an n-dimensional Boolean hypercube, our algorithm CubeProbeEst estimates the variation distance between P and Q within additive error using subcube conditional samples from P. Following the testing-via-learning paradigm, we also get a tester that distinguishes between the cases when P and Q are close or far in variation distance with high probability using subcube conditional samples. This estimation algorithm CubeProbeEst in the subcube conditioning sampling model helps us to design the first tester for self-reducible samplers. The correctness of the tester is formally proved. Moreover, we implement CubeProbeEst to test the quality of three samplers for sampling linear extensions in posets.