We design and analyze algorithms for online linear optimization that have
optimal regret and at the same time do not need to know any upper or lower
bounds on the norm of the loss vectors. Our algorithms are instances of the
Follow the Regularized Leader (FTRL) and Mirror Descent (MD) meta-algorithms...
We achieve adaptiveness to the norms of the loss vectors by scale invariance,
i.e., our algorithms make exactly the same decisions if the sequence of loss
vectors is multiplied by any positive constant. The algorithm based on FTRL
works for any decision set, bounded or unbounded. For unbounded decisions sets,
this is the first adaptive algorithm for online linear optimization with a
non-vacuous regret bound. In contrast, we show lower bounds on scale-free
algorithms based on MD on unbounded domains.