Reviews: Online Forecasting of Total-Variation-bounded Sequences

Update after reading the authors' response: The authors' response answered my questions well. One limitation that I missed in my initial review is that the coordinates of the parameter theta (the true signal sequence) are not assumed to be bounded; the only assumption is that theta lies in a total variation ball. This means that the only bound on these coordinates is through the bound C on the total variation of the sequence. Hence, the dependence on the bound B on the sequence (L-infty norm of theta) is implicit and replaced by the worst-case upper bound C, which leads to a dependence on C 2 instead of B*C on the intersection of those balls. I think that this limitation should be addressed, given that in the online learning literature it is more customary to provide the explicit dependence on B. To be specific, this would entail assuming that theta lies in the intersection of a TV ball of radius C and an L-infinity ball of radius B (where both B and C can be assumed to be known, though adaptivity to those can be considered), and providing matching upper and lower bound over this class.