  We consider the setting of online logistic regression and consider the regret
with respect to the 2-ball of radius B. It is known (see [Hazan et al., 2014])
that any proper algorithm which has logarithmic regret in the number of samples
(denoted n) necessarily suffers an exponential multiplicative constant in B. In
this work, we design an efficient improper algorithm that avoids this
exponential constant while preserving a logarithmic regret. Indeed, [Foster et
al., 2018] showed that the lower bound does not apply to improper algorithms
and proposed a strategy based on exponential weights with prohibitive
computational complexity. Our new algorithm based on regularized empirical risk
minimization with surrogate losses satisfies a regret scaling as O(B log(Bn))
with a per-round time-complexity of order O(d^2).
