Self-Normalized Martingales and Uniform Regret Bounds for Linear Regression

ArXi:2605.01628v1 Announce Type: cross Self-normalized martingale inequalities lie at the heart of confidence ellipsoids for online least squares and, broadly, many bandit and reinforcement-learning results. Yet existing vector and scalar results typically rely on bounded covariates and an explicit regularization matrix, producing bounds that are \emph{not scale-invariant}: although the self-normalized quantity is scale-invariant by definition, its standard upper bounds are not. We characterize when scale-invariant upper bounds on self-normalized martingales are possible.