  We consider online learning with linear models, where the algorithm predicts
on sequentially revealed instances (feature vectors), and is compared against
the best linear function (comparator) in hindsight. Popular algorithms in this
framework, such as Online Gradient Descent (OGD), have parameters (learning
rates), which ideally should be tuned based on the scales of the features and
the optimal comparator, but these quantities only become available at the end
of the learning process. In this paper, we resolve the tuning problem by
proposing online algorithms making predictions which are invariant under
arbitrary rescaling of the features. The algorithms have no parameters to tune,
do not require any prior knowledge on the scale of the instances or the
comparator, and achieve regret bounds matching (up to a logarithmic factor)
that of OGD with optimally tuned separate learning rates per dimension, while
retaining comparable runtime performance.
