Scale-invariant unconstrained online learning

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.