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Exploring the Gap Between Tolerant and Non-Tolerant Distribution Testing

Authors Sourav Chakraborty, Eldar Fischer, Arijit Ghosh, Gopinath Mishra, Sayantan Sen

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Author Details

Sourav Chakraborty
  • Indian Statistical Institute, Kolkata, India
Eldar Fischer
  • Technion - Israel Institute of Technology, Haifa, Israel
Arijit Ghosh
  • Indian Statistical Institute, Kolkata, India
Gopinath Mishra
  • University of Warwick, Coventry, UK
Sayantan Sen
  • Indian Statistical Institute, Kolkata, India

Funding

Gopinath Mishra’s research is supported in part by the Centre for Discrete Mathematics and its Applications (DIMAP) and by EPSRC award EP/V01305X/1.

Acknowledgements

The authors would like to thank the reviewers of RANDOM 2022 for their valuable suggestions which improved the presentation of the paper.

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Sourav Chakraborty, Eldar Fischer, Arijit Ghosh, Gopinath Mishra, and Sayantan Sen. Exploring the Gap Between Tolerant and Non-Tolerant Distribution Testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 27:1-27:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.27

Abstract

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 in some distance measure from satisfying it. 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 between the sample complexities of non-tolerant testing of distributions and their tolerant testing counterparts. When limiting our scope to label-invariant (symmetric) properties of distributions, we prove that the gap is at most quadratic, ignoring poly-logarithmic factors. 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, that is, the probability mass of the distribution is sufficiently spread out, then it cannot be non-tolerantly tested with o(√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 𝒪(n) many samples. Being non-concentrated is a strong requirement on properties, as we also prove a close to linear lower bound against their tolerant tests.
Apart from the case where the distribution is non-concentrated, we also show 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 𝒪(s) many samples. The learning procedure adapts to the input, and works without knowing s in advance.

Subject Classification

ACM Subject Classification
  • Theory of computation → Streaming, sublinear and near linear time algorithms
Keywords
  • Distribution Testing
  • Tolerant Testing
  • Non-tolerant Testing
  • Sample Complexity

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