Optimal Algorithms for Testing Closeness of Discrete Distributions (1308.3946v1)

Abstract: We study the question of closeness testing for two discrete distributions. More precisely, given samples from two distributions $p$ and $q$ over an $n$-element set, we wish to distinguish whether $p=q$ versus $p$ is at least $\eps$-far from $q$, in either $\ell_1$ or $\ell_2$ distance. Batu et al. gave the first sub-linear time algorithms for these problems, which matched the lower bounds of Valiant up to a logarithmic factor in $n$, and a polynomial factor of $\eps.$ In this work, we present simple (and new) testers for both the $\ell_1$ and $\ell_2$ settings, with sample complexity that is information-theoretically optimal, to constant factors, both in the dependence on $n$, and the dependence on $\eps$; for the $\ell_1$ testing problem we establish that the sample complexity is $\Theta(\max{n^{2/3}/\eps^{4/3}, n^{1/2}/\eps^2 }).$

• The paper presents novel algorithms that achieve optimal sample complexity for testing the closeness of discrete distributions in ℓ1 and ℓ2 norms.
• It employs techniques like Poissonization and variance bounding to simplify analysis and rigorously validate performance guarantees.
• The results have practical implications in data analysis and machine learning by enabling efficient testing with sub-linear sample sizes.

Optimal Algorithms for Testing Closeness of Discrete Distributions: A Summary

The paper "Optimal Algorithms for Testing Closeness of Discrete Distributions" examines the problem of determining whether two discrete probability distributions, $p$ and $q$, are statistically close or significantly different using as few samples as possible. Specifically, the work focuses on two common distance measures: $\ell_1$ and $\ell_2$.

Problem Context

Closeness testing is a well-established problem in statistics, with applications spanning across numerous domains such as data analysis, graph testing, and machine learning. Given samples from probability distributions $p$ and $q$ over an $n$-element set, the goal is to distinguish whether $p = q$ or whether $p$ is $\epsilon$-far from $q$ in terms of the $\ell_1$ or $\ell_2$ norm.

Key Contributions and Results

1. Optimal Sample Complexity for $\ell_1$ Testing: The authors present a novel and simplified algorithm for $\ell_1$ closeness testing. They establish that the sample complexity is $$\Theta(\max\{n^{2/3}/\epsilon^{4/3}, n^{1/2}/\epsilon^2 \})$$, which is proven to be optimal. This resolves a previously open question regarding the tight bounds for the problem, particularly in its dependence on the $\epsilon$ parameter.
2. Robust $\ell_2$ Closeness Testing: A robust algorithm is developed for $\ell_2$ testing, with a focus on tightly estimating the $\ell_2$ distance between $p$ and $q$. The sample complexity needed to reach a specific error tolerance in $\ell_2$ distance is shown to be $O(\sqrt{b}/\epsilon^2)$, where $b$ bounds the squared $\ell_2$ norm of the distributions. This complexity is also shown to be optimal for robust testing.

Methodological Approach

The authors employ a combination of theoretical insights and algorithmic advanced approaches. They leverage existing techniques such as the Poissonization approach to simplify the distribution of the number of observations, reducing the complexity of their algorithms' analysis. Their methodology includes careful bounding of variances, application of Cauchy-Schwarz, and Chebyshev's inequalities, ensuring that their algorithms perform within the expected bounds with high probability.

Implications and Future Directions

The results have significant practical implications in fields where efficiently distinguishing distributional closeness is essential. This includes applications in big data environments, where sub-linear sample complexity is crucial to manage computation costs effectively. Theoretically, these results also open pathways to exploring similar optimal methodologies across related problems, such as testing symmetries or graph properties.

Future research can explore extending these results to deal with more complex structured distributions or adapting these methodologies for online learning scenarios. Furthermore, understanding the role of additional data structures or parallel computation strategies could enhance the practical applicability of the work.

In summary, this paper contributes valuable insights and tools to the field of distribution property testing, offering optimal, efficient solutions to the fundamental problem of closeness testing across discrete distributions.