Section: New Results
Uniform regret bounds over for the sequential linear regression problem with the square loss
In  we consider the setting of online linear regression for arbitrary deterministic sequences, with the square loss. We are interested in obtaining regret bounds that hold uniformly over all vectors . When the feature sequence is known at the beginning of the game, they provided closed-form regret bounds of , where is the number of rounds and is a bound on the observations. Instead, we derive bounds with an optimal constant of 1 in front of the term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of for any individual sequence of features and bounded observations. All our algorithms are variants of the online nonlinear ridge regression forecaster, either with a data-dependent regularization or with almost no regularization.