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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(?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.
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