# Variational Tail Bounds for Norms of Random Vectors and Matrices

arXiv: https://arxiv.org/abs/2503.17300 · HTML: https://mlsanity.com/paper/2503.17300/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 51 references that are in the map.
- First posted on arXiv 21 Mar 2025.

## Builds on (most cited first, 3 of 3)

- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Non-Vacuous Generalization Bounds at the ImageNet Scale: A PAC-Bayesian Compression Approach](https://mlsanity.com/paper/1804.05862/) (2018)
- [On the Injective Norm of Sums of Random Tensors and the Moments of Gaussian Chaoses](https://mlsanity.com/paper/2503.10580/) (2025)

## Abstract (arXiv)

We propose a variational tail bound for norms of random vectors and matrices under moment assumptions on their one-dimensional marginals. A simplified version of the bound that parametrizes the ``aggregating distribution'' using a certain pushforward of the Gaussian distribution is also provided. We apply the proposed method to reproduce some of the well-known bounds on norms of Gaussian random vectors, and also obtain dimension-free tail bounds for the Euclidean norm of random vectors with arbitrary moment profiles. Furthermore, we reproduce a dimension-free concentration inequality for sum of independent and identically distributed positive semidefinite matrices with sub-exponential marginals, and obtain a concentration inequality for the sample covariance matrix of sub-exponential random vectors. We also obtain a tail bound for the operator norm of a random matrix series whose random coefficients may have arbitrary moment profiles. Furthermore, we use coupling to formulate an abstraction of the proposed approach that applies more broadly. As a corollary, we derive a PAC-Bayesian-style bound in terms of a certain combination of the KL and R'enyi divergences between the prior and posterior distributions.