McDiarmid's Inequality Extended to Dependent Random Variables via Entropy Tensorization
Researchers demonstrate that approximate tensorization of entropy (ATE) provides a powerful framework for deriving McDiarmid's inequality under dependence, extending classical concentration results to dependent settings. The work connects ATE to McDiarmid's inequality through the Entropy Method and applies these results to non-isotropic Gaussian vectors, log-concave measures, and graph theory problems. This development makes dependent concentration inequalities more accessible and practical for applications in statistics, learning theory, and theoretical computer science.
The paper argues that dependent versions of McDiarmid's inequality—a fundamental concentration inequality in probability theory—deserve greater attention in mathematical statistics and theoretical computer science. The authors establish that approximate tensorization of entropy implies McDiarmid's inequality via the Entropy Method, then derive McDiarmid's inequality for non-isotropic Gaussian random vectors with constants depending on the condition number of the covariance matrix. They obtain this result both through stochastic localization and by generalizing prior work on ATE for the Gibbs sampler to strongly log-concave and log-smooth measures. The paper applies these concentration inequalities to several problems: resolving a question about concentration of sign(X), analyzing Erdős-Rényi random graphs under dependence, and proving a Dvoretzky-Kiefer-Wolfowitz-type inequality for non-i.i.d. observations that achieves the optimal 1/√n convergence rate under weak dependence, improving upon prior results that only achieved n^(-1/3) rates.
What's missing
The paper does not discuss computational aspects of applying these concentration inequalities in practice, nor does it provide empirical validation of the theoretical bounds through simulations or real-world applications.
What different sources said
- arXiv stat.MLCenter
On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy
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