McDiarmid's Inequality Extended to Dependent Random Variables via Entropy Tensorization
Researchers have derived new concentration inequalities for dependent random variables by linking approximate tensorization of entropy (ATE) to McDiarmid's inequality via the Entropy Method. The work applies these results to Gaussian random vectors, Erdős-Rényi graphs, and empirical distribution functions, and improves a 2010 Dvoretzky-Kiefer-Wolfowitz-type bound from a convergence rate of n⁻¹/³ to the optimal 1/√n under weak dependence. This matters because concentration inequalities for dependent data are foundational tools in statistics, learning theory, and theoretical computer science, yet dependent versions have been underutilized.
A new preprint posted to arXiv argues that dependent versions of McDiarmid's inequality are underutilized in mathematical statistics, machine learning theory, and theoretical computer science, and proceeds to substantially advance their theoretical foundations. The authors establish that approximate tensorization of entropy (ATE) implies McDiarmid's inequality through the Entropy Method, then derive a version of the inequality for non-isotropic Gaussian random vectors whose constant scales with the condition number of the covariance matrix. This ATE result is obtained independently via stochastic localization and is also connected to a more general ATE for the Gibbs sampler due to Ascolani et al. (2026), which extends McDiarmid-like concentration to strongly log-concave and log-smooth probability measures. Among the applications, the authors resolve an open question on the concentration of the sign function of Gaussian vectors, analyze Erdős-Rényi random graphs under dependence, and prove a Dvoretzky-Kiefer-Wolfowitz-type inequality for observations from a joint measure satisfying ATE with continuous marginal CDFs. Notably, for strongly log-concave and log-smooth measures, this last result improves upon a prior bound by Bobkov and Götze (2010) by recovering the statistically expected 1/√n convergence rate rather than the slower n⁻¹/³ rate previously established for non-i.i.d. observations.
What's missing
The paper is a preprint and has not yet undergone peer review. Empirical validation or simulation studies illustrating the tightness of the new bounds in practice are not included.
What different sources said
- arXiv stat.MLCenter
On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy
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