# Sharp Matrix Empirical Bernstein Inequalities

arXiv: https://arxiv.org/abs/2411.09516 · HTML: https://mlsanity.com/paper/2411.09516/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 26 references that are in the map.
- First posted on arXiv 14 Nov 2024.

## Builds on (most cited first, 11 of 11)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Tight Concentrations and Confidence Sequences from the Regret of Universal Portfolio](https://mlsanity.com/paper/2110.14099/) (2021)
- [PAC-Bayes Un-Expected Bernstein Inequality](https://mlsanity.com/paper/1905.13367/) (2019)
- [Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities](https://mlsanity.com/paper/2304.02611/) (2023)
- [Tighter PAC-Bayes Bounds Through Coin-Betting](https://mlsanity.com/paper/2302.05829/) (2023)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [Risk-Monotonicity in Statistical Learning](https://mlsanity.com/paper/2011.14126/) (2020)
- [Time-Uniform Confidence Spheres for Means of Random Vectors](https://mlsanity.com/paper/2311.08168/) (2023)

## Built on by (most cited first, 2 of 2)

- [A variational approach to dimension-free self-normalized concentration](https://mlsanity.com/paper/2508.06483/) (2025)
- [Cheap to Draw, Expensive to Trust: Certifying Test-Time Scaling Curves](https://mlsanity.com/paper/2609.40190/) (2026)

## Abstract (arXiv)

We present two sharp, closed-form empirical Bernstein inequalities for symmetric random matrices with bounded eigenvalues. By sharp, we mean that both inequalities adapt to the unknown variance in a tight manner: the deviation captured by the first-order 1/√n term asymptotically matches the matrix Bernstein inequality exactly, including constants, the latter requiring knowledge of the variance. Our first inequality holds for the sample mean of independent matrices, and our second inequality holds for a mean estimator under martingale dependence at stopping times.