Bernstein-type dimension-free concentration for self-normalised martingales

TL;DR

This paper presents a dimension-free Bernstein-type tail inequality for self-normalised martingales, enabling new confidence sequences and regret bounds in high-dimensional statistical models.

Contribution

It introduces a novel tail inequality that is dimension-free and applicable to self-normalised martingales, with applications to logistic regression and bandit problems.

Findings

Derived a dimension-free Bernstein-type tail inequality.

Established ellipsoidal confidence sequences for logistic regression.

Provided instance-adaptive regret bounds for Hilbert-armed logistic bandits.

Abstract

We introduce a dimension-free Bernstein-type tail inequality for self-normalised martingales, where the normalisation uses the predictable quadratic variation and the radius depends on the information gain of the observed covariance. As applications, we provide ellipsoidal confidence sequences for logistic regression with adaptively chosen Hilbert-valued covariates, and give instance-adaptive regret bounds for Hilbert-armed logistic bandits.