Uniform regret bounds over $R^d$ for the sequential linear regression problem with the square loss

Abstract: We consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in the aim set by Bartlett et al. (2015): obtain regret bounds that hold uniformly over all competitor vectors. When the feature sequence is known at the beginning of the game, they provided closed-form regret bounds of $2d B² \ln T + \mathcal{O}_T(1)$, where $T$ is the number of rounds and $B$ is a bound on the observations. Instead, we derive bounds with an optimal constant of $1$ in front of the $d B² \ln T$ term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of $d B² \ln T$ for any individual sequence of features and bounded observations. All our algorithms are variants of the online non-linear ridge regression forecaster, either with a data-dependent regularization or with almost no regularization.