# On McDiarmid's Inequality under Dependence via Approximate Tensorization of Entropy

arXiv: https://arxiv.org/abs/2606.12720 · HTML: https://mlsanity.com/paper/2606.12720/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 68 references that are in the map.
- First posted on arXiv 10 Jun 2026.

## Builds on (most cited first, 14 of 14)

- [Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization](https://mlsanity.com/paper/2203.05093/) (2022)
- [Sampling, Diffusions, and Stochastic Localization](https://mlsanity.com/paper/2305.10690/) (2023)
- [Memorization and Optimization in Deep Neural Networks with Minimum Over-parameterization](https://mlsanity.com/paper/2205.10217/) (2022)
- [McDiarmid-Type Inequalities for Graph-Dependent Variables and Stability Bounds](https://mlsanity.com/paper/1909.02330/) (2019)
- [Eldan's Stochastic Localization and the KLS Conjecture: Isoperimetry, Concentration and Mixing](https://mlsanity.com/paper/1612.01507/) (2016)
- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion](https://mlsanity.com/paper/2011.02075/) (2020)
- [A Slightly Improved Bound for the KLS Constant](https://mlsanity.com/paper/2208.11644/) (2022)
- [An extension of McDiarmid's inequality](https://mlsanity.com/paper/1511.05240/) (2015)
- [On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization](https://mlsanity.com/paper/2103.07459/) (2021)
- [Trickle-Down in Localization Schemes and Applications](https://mlsanity.com/paper/2407.16104/) (2024)
- [Universality of Spectral Independence with Applications to Fast Mixing in Spin Glasses](https://mlsanity.com/paper/2307.10466/) (2023)
- [Perspectives on Stochastic Localization](https://mlsanity.com/paper/2510.04460/) (2025)
- [On the Subgaussianity of Quantized Linear Maps: An AI-Assisted Note](https://mlsanity.com/paper/2605.27563/) (2026)

## Abstract (arXiv)

We argue that dependent versions of McDiarmid's inequality are a useful but underutilized tool in mathematical statistics, learning theory and theoretical computer science. To make this point, we first highlight that approximate tensorization of entropy (ATE) implies McDiarmid's via the Entropy Method. Second, we derive McDiarmid's inequality for non-isotropic Gaussian random vectors X ~ mathcal N(μ, Sigma) through ATE with a constant of the order of the condition number of Sigma. We both independently obtain this ATE through a simple application of stochastic localization and also discuss how a more general ATE for the Gibbs sampler due to Ascolani et al., 2026 generalizes McDiarmid's-like concentration to strongly log-concave and log-smooth probability measures. We then apply the resulting concentration inequalities to resolve a question on the concentration of operatornamesign(X) posed by Simone Bombari, investigate ErdHos-R'enyi graphs under dependence and prove a Dvoretzky-Kiefer-Wolfowitz-type inequality for observations from a joint measure fulfilling ATE and continuous marginal CDFs. For the class of strongly log-concave and log-smooth measures, this result improves upon a prior Dvoretzky-Kiefer-Wolfowitz-type inequality for non-i.i.d. observations due to Bobkov and G"otze, 2010, by establishing the expected 1/√n-rate of convergence under weak dependence instead of n-1/3.