# Optimal Testing of Discrete Distributions with High Probability

arXiv: https://arxiv.org/abs/2009.06540 · HTML: https://mlsanity.com/paper/2009.06540/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2020.

## Built on by (most cited first, 15 of 15)

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- [Better Private Distribution Testing by Leveraging Unverified Auxiliary Data](https://mlsanity.com/paper/2503.14709/) (2025)
- [Replicable Distribution Testing](https://mlsanity.com/paper/2507.02814/) (2025)
- [Sharp Constants in Uniformity Testing via the Huber Statistic](https://mlsanity.com/paper/2206.10722/) (2022)
- [Evaluation of Categorical Generative Models -- Bridging the Gap Between Real and Synthetic Data](https://mlsanity.com/paper/2210.16405/) (2022)
- [Testing Credibility of Public and Private Surveys through the Lens of Regression](https://mlsanity.com/paper/2410.05458/) (2024)
- [Efficient Sample-optimal Learning of Gaussian Tree Models via Sample-optimal Testing of Gaussian Mutual Information](https://mlsanity.com/paper/2411.11516/) (2024)
- [Sample Complexity of Nonparametric Closeness Testing for Continuous Distributions and Its Application to Causal Discovery with Hidden Confounding](https://mlsanity.com/paper/2503.07475/) (2025)
- [Entropy Equivalence Testing](https://mlsanity.com/paper/2605.23225/) (2026)

## Abstract (arXiv)

We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property P, and parameters 0< ε, δ <1, we want to distinguish em with probability at least 1-δ whether these distributions satisfy P or are ε-far from P in total variation distance. Most prior work in distribution testing studied the constant confidence case (corresponding to δ = Omega(1)), and provided sample-optimal testers for a range of properties. While one can always boost the confidence probability of any such tester by black-box amplification, this generic boosting method typically leads to sub-optimal sample bounds. Here we study the following broad question: For a given property P, can we em characterize the sample complexity of testing P as a function of all relevant problem parameters, including the error probability δ? Prior to this work, uniformity testing was the only statistical task whose sample complexity had been characterized in this setting. As our main results, we provide the first algorithms for closeness and independence testing that are sample-optimal, within constant factors, as a function of all relevant parameters. We also show matching information-theoretic lower bounds on the sample complexity of these problems. Our techniques naturally extend to give optimal testers for related problems. To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions and testing closeness with unequal sized samples.