applications in both theory and practice; 2) our black-box approach provides a much more intuitive understanding

We thank all reviewers for their valuable comments. This is admittedly true from a theoretical viewpoint. Therefore, we believe that the significance of our results goes beyond the theoretical improvement of regret bounds. We will add more discussion on this in the next version of our paper, as suggested by the reviewer. For the bandit setting, again there is no known lower bound.