Known algorithms applied to online logistic regression on a feasible set of L2
diameter D achieve regret bounds like O(eD log T) in one dimension, but
we show a bound of O(sqrt(D) + log T) is possible in a binary 1-dimensional
problem. Thus, we pose the following question: Is it possible to achieve a regret
bound for online logistic regression that is O(poly(D)log(T))? Even if this is not
possible in general, it would be interesting to have a bound that reduces to our
bound in the one-dimensional case.