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 B^2 ln T + O(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 B^2 ln T term. In the case of sequentially revealed features, we also derive an asymptotic regret bound of d B^2 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.