# A variational approach to dimension-free self-normalized concentration

arXiv: https://arxiv.org/abs/2508.06483 · HTML: https://mlsanity.com/paper/2508.06483/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 73 references that are in the map.
- First posted on arXiv 8 Aug 2025.

## Builds on (most cited first, 21 of 21)

- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [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)
- [Nearly Minimax Optimal Reinforcement Learning for Linear Mixture Markov Decision Processes](https://mlsanity.com/paper/2012.08507/) (2020)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Improved Optimistic Algorithms for Logistic Bandits](https://mlsanity.com/paper/2002.07530/) (2020)
- [Sub-sampled Cubic Regularization for Non-convex Optimization](https://mlsanity.com/paper/1705.05933/) (2017)
- [On the Sublinear Regret of GP-UCB](https://mlsanity.com/paper/2307.07539/) (2023)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)
- [A Tutorial on Concentration Bounds for System Identification](https://mlsanity.com/paper/1906.11395/) (2019)
- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
- [Improved Algorithms for Stochastic Linear Bandits Using Tail Bounds for Martingale Mixtures](https://mlsanity.com/paper/2309.14298/) (2023)
- [Time-Uniform Self-Normalized Concentration for Vector-Valued Processes](https://mlsanity.com/paper/2310.09100/) (2023)
- [Confidence Estimation via Sequential Likelihood Mixing](https://mlsanity.com/paper/2502.14689/) (2025)
- [Time-Uniform Confidence Spheres for Means of Random Vectors](https://mlsanity.com/paper/2311.08168/) (2023)
- [Mean Estimation in Banach Spaces Under Infinite Variance and Martingale Dependence](https://mlsanity.com/paper/2411.11271/) (2024)
- [Generalized Kernelized Bandits: A Novel Self-Normalized Bernstein-Like Dimension-Free Inequality and Regret Bounds](https://mlsanity.com/paper/2508.01681/) (2025)
- [Achieving Constant Regret in Linear Markov Decision Processes](https://mlsanity.com/paper/2404.10745/) (2024)
- [Sharp Matrix Empirical Bernstein Inequalities](https://mlsanity.com/paper/2411.09516/) (2024)
- [Dimension-free Structured Covariance Estimation](https://mlsanity.com/paper/2402.10032/) (2024)

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

- [Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity](https://mlsanity.com/paper/2511.03606/) (2025)
- [Sequential operator learning under dependent data](https://mlsanity.com/paper/2608.24426/) (2026)

## Abstract (arXiv)

We study the self-normalized concentration of vector-valued stochastic processes. We focus on bounds for "sub-psi" processes, a well-known and quite general class that encompasses a wide variety of well-known tail conditions (including sub-exponential, sub-Gaussian, sub-gamma, sub-Poisson, and several heavy-tailed settings without a moment generating function such as symmetric or bounded 2nd or 3rd moments). Our results recover and generalize the influential bound of de la Pe~na et al. [20] (proved again in Abbasi-Yadkori et al. [2]) in the sub-Gaussian case. Further, we fill a gap in the literature between determinant-based bounds and more recent bounds based on condition numbers. As applications we prove a Bernstein inequality for random vectors satisfying a moment condition (a more general condition than boundedness), and also provide the first dimension-free self-normalized empirical Bernstein inequality. Our techniques are based on the variational (PAC-Bayes) approach to concentration.