Scale-Free Online Learning
This work addresses the need for adaptive online learning algorithms that eliminate the requirement for norm bounds, with incremental improvements for unbounded decision sets in optimization.
The authors tackled the problem of online linear optimization without requiring prior knowledge of loss vector norms, achieving optimal regret through scale-invariant algorithms based on Follow the Regularized Leader (FTRL) and Mirror Descent (MD). They developed the first adaptive algorithm with non-vacuous regret bounds for unbounded decision sets using FTRL, while showing lower bounds for scale-free MD algorithms on unbounded domains.
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.