Self-Normalized Martingales and Uniform Regret Bounds for Linear Regression
Self-normalized martingale inequalities lie at the heart of confidence ellipsoids for online least squares and, more 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. Without further assumptions, we prove that nontrivial scale-invariant bounds exist only in dimension $d=1$; moreover, in $d=1$ we obtain $O(\log T)$ sca...
| Authors | Fan Chen, Jian Qian, Alexander Rakhlin et al. |
|---|---|
| Primary category | stat.ML |
| Published | 2 May 2026 |
| Last updated at source | 2 May 2026 |
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