Self-Normalized Martingales and Uniform Regret Bounds for Linear Regression
For researchers in online learning and bandit theory, the paper resolves a fundamental open problem about the limits of scale-invariant regret bounds, providing a clear dichotomy between one-dimensional and higher-dimensional settings.
The paper characterizes when scale-invariant self-normalized martingale bounds exist, proving they are only possible in dimension d=1 (with O(log T) bounds) and impossible for d>1 without additional assumptions. This resolves an open question about doubly-uniform regret in online linear regression, showing sublinear regret is achievable in d=1 but impossible for d>1, unless a smoothness condition is imposed.
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 $O(\log T)$ scale-invariant self-normalized bounds without any assumptions on the covariates. In contrast, for $d>1$ we show that no nontrivial scale-invariant bound can hold in full generality. We then connect this dichotomy to \emph{doubly-uniform} regret in online linear regression (i.e., regret bounds that are simultaneously independent of the covariate scale and the comparator norm) and use it to resolve the open question of Gaillard, Gerchinovitz, Huard, and Stoltz, \emph{``Uniform regret bounds over $\mathbb{R}^d$ for the sequential linear regression problem with the square loss''} (ALT 2019): in $d=1$ we give an explicit algorithm with $O(\log T)$ doubly-uniform regret, whereas for $d>1$ sublinear doubly-uniform regret is impossible. Finally, under a natural \emph{smoothness} condition (bounded Radon--Nikodym derivatives of the conditional covariate laws with respect to a fixed base measure), we recover sublinear regret for $d>1$ without bounded covariates and derive a self-normalized concentration inequality free of the usual regularization penalties, yielding arguably a first natural scale-invariant bound for adaptive, non-i.i.d. vector martingales.