A New Approach for Testing Properties of Discrete Distributions (1601.05557v2)

Abstract: In this work, we give a novel general approach for distribution testing. We describe two techniques: our first technique gives sample-optimal testers, while our second technique gives matching sample lower bounds. As a consequence, we resolve the sample complexity of a wide variety of testing problems. Our upper bounds are obtained via a modular reduction-based approach. Our approach yields optimal testers for numerous problems by using a standard $\ell_2$-identity tester as a black-box. Using this recipe, we obtain simple estimators for a wide range of problems, encompassing most problems previously studied in the TCS literature, namely: (1) identity testing to a fixed distribution, (2) closeness testing between two unknown distributions (with equal/unequal sample sizes), (3) independence testing (in any number of dimensions), (4) closeness testing for collections of distributions, and (5) testing histograms. For all of these problems, our testers are sample-optimal, up to constant factors. With the exception of (1), ours are the {\em first sample-optimal testers for the corresponding problems.} Moreover, our estimators are significantly simpler to state and analyze compared to previous results. As an application of our reduction-based technique, we obtain the first {\em nearly instance-optimal} algorithm for testing equivalence between two {\em unknown} distributions. Moreover, our technique naturally generalizes to other metrics beyond the $\ell_1$-distance. Our lower bounds are obtained via a direct information-theoretic approach: Given a candidate hard instance, our proof proceeds by bounding the mutual information between appropriate random variables. While this is a classical method in information theory, prior to our work, it had not been used in distribution property testing.

• The paper introduces a reduction-based framework that delivers sample-optimal testers with matching lower bounds for various discrete distribution testing problems.
• It transforms complex tasks like identity, closeness, independence, and histogram testing into a simplified ℓ2-identity testing problem.
• The methodology leverages information theory to provide tight sample complexity bounds, offering efficient insights for machine learning and statistical applications.

A New Approach for Testing Properties of Discrete Distributions

This paper by Diakonikolas and Kane presents a sophisticated framework for distribution property testing that achieves sample complexity bounds that are both upper and lower, demonstrating efficiency and optimality across a range of detection problems. The authors introduce a generalized approach applicable to various scenarios in the paper of determining global properties of distributions using sample data.

Core Contributions

The paper outlines two primary techniques. The first technique establishes sample-optimal testers, and the second provides matching sample lower bounds, effectively determining the sample complexity for a diverse array of testing problems. The methodologies are applicable to:

1. Identity Testing: Verification against a fixed distribution.
2. Closeness Testing: For both identical and distinct sample sizes.
3. Independence Testing: Across multiple dimensions.
4. Testing Collections: Involving multiple distributions.
5. Histogram Testing: Evaluating piecewise distribution behavior.

The results yield significant gains in understanding the sample complexities for these problems, with the authors providing the first sample-optimal testers for several of them.

Methodological Advancements

The novel reduction-based framework introduced here transforms complex distribution testing problems into simpler ones via modular reductions. This is built around a reduction to $\ell_2$-identity testing, allowing for the construction of sample-optimal estimators that are not only methodologically simple but provide improvements over prior techniques.

1. Reduction-based Testing:
   • The approach uses a basic $\ell_2$-identity tester to evaluate $\ell_1$-distance between distributions. These transformations allow for sample-efficient implementations.
   • The framework simplifies the formulation and analysis of testers by reducing to a specific critical $\ell_2$ problem.
2. Information-Theoretic Lower Bounds:
   • Lower bounds are established using a classical method that involves bounding mutual information, providing tight sample complexity bounds for the listed problems.
   • This reliance on information theory contrasts previous methods which dealt with symmetric properties using moment-matching or the birthday paradox.

Implications and Future Directions

The implications of this work extend to both theory and practical applications within machine learning and statistics, potentially aiding in tasks involving distribution fits and hypothesis testing where sample efficiency is crucial. In particular, researchers concerned with statistical inferencing and property testing in high-dimensional spaces may find these methods compelling due to their optimal resource utilization.

Future work could see these methods further adapted to handle a wider range of divergence measures beyond $\ell_1$, as some results here already extend to Hellinger distance. Moreover, expanding this framework to handle dynamic distributions (those varying over time or contexts) while maintaining sample optimality can open new avenues in real-time statistical analysis.

Conclusion

This research lays a robust groundwork for advancing the efficiency of discrete distribution testing. The methodologies introduced could serve as a baseline for theoretical exploration and practical applications in statistical learning and information theory. The well-defined reduction-based approach and information-theoretic underpinnings position this paper as a reference point for subsequent investigations into distribution property testing.