Concentration of measure without independence: a unified approach via the martingale method (1602.00721v3)

Abstract: The concentration of measure phenomenon may be summarized as follows: a function of many weakly dependent random variables that is not too sensitive to any of its individual arguments will tend to take values very close to its expectation. This phenomenon is most completely understood when the arguments are mutually independent random variables, and there exist several powerful complementary methods for proving concentration inequalities, such as the martingale method, the entropy method, and the method of transportation inequalities. The setting of dependent arguments is much less well understood. This chapter focuses on the martingale method for deriving concentration inequalities without independence assumptions. In particular, we use the machinery of so-called Wasserstein matrices to show that the Azuma-Hoeffding concentration inequality for martingales with almost surely bounded differences, when applied in a sufficiently abstract setting, is powerful enough to recover and sharpen several known concentration results for nonproduct measures. Wasserstein matrices provide a natural formalism for capturing the interplay between the metric and the probabilistic structures, which is fundamental to the concentration phenomenon.

• The paper extends classical concentration inequalities by applying a martingale method to weakly dependent random variables.
• It introduces Wasserstein matrices to capture the interplay between metric and probabilistic structures, refining known results.
• The approach offers practical implications for developing robust algorithms in network analysis, reinforcement learning, and related fields.

Concentration of Measure without Independence: A Unified Approach via the Martingale Method

Overview

The paper by Aryeh Kontorovich and Maxim Raginsky explores a sophisticated approach to concentration of measure phenomena without relying on the assumption of independence between variables. This work primarily utilizes the martingale method and introduces the concept of Wasserstein matrices to tackle concentration inequalities in settings where variables exhibit weak dependencies, thus extending the scope of classical concentration results traditionally confined to independent scenarios.

Key Contributions

1. Martingale Method Extension: The research presents a framework leveraging the martingale method to address concentration inequalities involving weakly dependent random variables. This method offers a robust alternative to traditional techniques such as the entropy method and transportation inequalities.
2. Wasserstein Matrices: The authors introduce the use of Wasserstein matrices as a formalism to capture the interplay between metric and probabilistic structures inherent to the concentration of measure. These matrices allow for a more nuanced analysis of dependencies and their influence on concentration properties.
3. Recovery and Refinement of Known Results: By applying their framework in an abstract setting, the authors not only recover existing concentration results applicable to non-product measures but also refine them, demonstrating the power of their method in recapturing well-known inequalities under weaker assumptions.

Discussion

The concentration of measure phenomenon, vital in high-dimensional statistics and probability, finds its complete understanding in independent scenarios. However, this research bridges the gap for dependent variables by utilizing Wasserstein matrices, which provide valuable estimates of local variability and facilitate the derivation of general purpose concentration bounds.

Practical and Theoretical Implications

• Theoretical Implications: The theoretical contributions, particularly the use of Wasserstein matrices, provide a uniform approach to assessing concentration inequalities across a broader range of dependence structures, potentially influencing various fields like statistical mechanics and information theory.
• Practical Implications: These insights could be pivotal in developing better algorithms in areas where dependencies are inevitable, such as network analysis, reinforcement learning, and certain machine learning tasks, enhancing the robustness and reliability of algorithms under realistic conditions.

Future Directions

The research opens several avenues for further exploration:

• Broader Application of Wasserstein Matrices: There is potential to leverage Wasserstein matrices beyond measure concentration, possibly applying the concept in areas such as optimal transport and distributional robustness.
• Integration with Other Methods: Combining the martingale method with other approaches like Stein’s method or spectral techniques could yield even sharper concentration inequalities.
• Empirical Evaluation: Empirically testing the theoretical findings in real-world datasets to validate the practical utility of the proposed method.

Conclusion

The paper provides a comprehensive and technically enriching framework for addressing concentration of measure issues in the absence of independence. By extending traditional methodologies through the innovative use of Wasserstein matrices and martingales, it sets a foundation for future research efforts aimed at understanding complex dependencies in probabilistic systems. This advancement not only enriches the theoretical landscape but also broadens the applicability of concentration inequalities in various scientific domains.