  We study Online Convex Optimization in the unbounded setting where neither
predictions nor gradient are constrained. The goal is to simultaneously adapt
to both the sequence of gradients and the comparator. We first develop
parameter-free and scale-free algorithms for a simplified setting with hints.
We present two versions: the first adapts to the squared norms of both
comparator and gradients separately using $O(d)$ time per round, the second
adapts to their squared inner products (which measure variance only in the
comparator direction) in time $O(d^3)$ per round. We then generalize two prior
reductions to the unbounded setting; one to not need hints, and a second to
deal with the range ratio problem (which already arises in prior work). We
discuss their optimality in light of prior and new lower bounds. We apply our
methods to obtain sharper regret bounds for scale-invariant online prediction
with linear models.
