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On integral probability metrics, φ-divergences and binary classification (0901.2698v4)

Published 18 Jan 2009 in cs.IT and math.IT

Abstract: A class of distance measures on probabilities -- the integral probability metrics (IPMs) -- is addressed: these include the Wasserstein distance, Dudley metric, and Maximum Mean Discrepancy. IPMs have thus far mostly been used in more abstract settings, for instance as theoretical tools in mass transportation problems, and in metrizing the weak topology on the set of all Borel probability measures defined on a metric space. Practical applications of IPMs are less common, with some exceptions in the kernel machines literature. The present work contributes a number of novel properties of IPMs, which should contribute to making IPMs more widely used in practice, for instance in areas where $\phi$-divergences are currently popular. First, to understand the relation between IPMs and $\phi$-divergences, the necessary and sufficient conditions under which these classes intersect are derived: the total variation distance is shown to be the only non-trivial $\phi$-divergence that is also an IPM. This shows that IPMs are essentially different from $\phi$-divergences. Second, empirical estimates of several IPMs from finite i.i.d. samples are obtained, and their consistency and convergence rates are analyzed. These estimators are shown to be easily computable, with better rates of convergence than estimators of $\phi$-divergences. Third, a novel interpretation is provided for IPMs by relating them to binary classification, where it is shown that the IPM between class-conditional distributions is the negative of the optimal risk associated with a binary classifier. In addition, the smoothness of an appropriate binary classifier is proved to be inversely related to the distance between the class-conditional distributions, measured in terms of an IPM.

Citations (203)

Summary

  • The paper establishes that the total variation distance is the only non-trivial φ-divergence qualifying as an IPM, clarifying their fundamental differences.
  • It derives empirical estimators for key IPMs and demonstrates superior convergence rates in high-dimensional settings compared to φ-divergence estimators.
  • The study connects IPMs to binary classification by showing how these metrics inform optimal risk assessments for Lipschitz classifiers.

An Overview of "On Integral Probability Metrics, ϕ\phi-Divergences and Binary Classification"

This paper examines Integral Probability Metrics (IPMs) as a framework for quantifying distances between probability measures. Notable IPM examples studied include the Wasserstein distance, Dudley metric, and Maximum Mean Discrepancy (MMD). These metrics have traditionally been employed in theoretical applications such as the mass transportation problem and as tools to metrize the weak topology on Borel probability measures. The research presented in this paper seeks to promote IPMs' practical utility, particularly in areas where ϕ\phi-divergences, such as the Kullback-Leibler (KL)-divergence, are currently predominant.

Key Contributions

  1. Relationship between IPMs and ϕ\phi-Divergences: This work investigates the conditions under which IPMs and ϕ\phi-divergences coincide, revealing that the total variation distance is the sole non-trivial ϕ\phi-divergence that is also an IPM. This finding highlights fundamental differences between IPMs and ϕ\phi-divergences, suggesting that the former may offer some distinct advantages, particularly in estimation scenarios.
  2. Empirical Estimation of IPMs: The authors derive empirical estimators for key IPMs from finite independent and identically distributed (i.i.d.) samples. These estimators are evaluated in terms of consistency and convergence rates, and it is established that they generally exhibit superior convergence behavior compared to estimators for ϕ\phi-divergences, particularly in high-dimensional settings.
  3. Connections to Binary Classification: The paper presents a novel interpretation of IPMs in the context of binary classification. It is shown that IPMs between class-conditional distributions can be related to the optimal risk associated with binary classifiers. For instance, the magnitude of the Wasserstein distance is inversely related to the smoothness of an appropriate Lipschitz classifier.

Practical and Theoretical Implications

The work suggests that IPMs could find widespread application in statistical inference contexts where distributional distance measures are needed. For instance, they could play a pivotal role in scenarios requiring robust distribution testing or classification tasks that hinge on accurately measuring distributional discrepancies. The demonstrated computability and strong convergence properties of IPM estimators mean they could be deployed more efficiently than their ϕ\phi-divergence counterparts in practical high-dimensional inference tasks.

Speculations on Future Developments

Future research could focus on exploring the use of IPMs beyond the existing scope, venturing into novel domains such as complex data structures or advanced machine learning algorithms. Given that the paper just begins to delineate the broader landscape of IPM applications, new theoretical developments related to IPM properties, such as exploring their intersection with Bregman divergences, appear promising.

Conclusion

This paper effectively broadens the theoretical foundation and potential application scope of IPMs. With insightful theoretical contributions and practical estimation methodologies, the research makes a compelling case for integrating IPMs into areas traditionally dominated by ϕ\phi-divergences. The paper, therefore, sets the stage for further investigations into the adoption of IPMs across a spectrum of probabilistic and statistical applications, paving the way for more comprehensive and versatile analysis tools in contemporary data-driven disciplines.

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