  We consider a variant of online convex optimization in which both the
instances (input vectors) and the comparator (weight vector) are unconstrained.
We exploit a natural scale invariance symmetry in our unconstrained setting:
the predictions of the optimal comparator are invariant under any linear
transformation of the instances. Our goal is to design online algorithms which
also enjoy this property, i.e. are scale-invariant. We start with the case of
coordinate-wise invariance, in which the individual coordinates (features) can
be arbitrarily rescaled. We give an algorithm, which achieves essentially
optimal regret bound in this setup, expressed by means of a coordinate-wise
scale-invariant norm of the comparator. We then study general invariance with
respect to arbitrary linear transformations. We first give a negative result,
showing that no algorithm can achieve a meaningful bound in terms of
scale-invariant norm of the comparator in the worst case. Next, we compliment
this result with a positive one, providing an algorithm which "almost" achieves
the desired bound, incurring only a logarithmic overhead in terms of the norm
of the instances.
