  The logistic loss function is often advocated in machine learning and
statistics as a smooth and strictly convex surrogate for the 0-1 loss. In this
paper we investigate the question of whether these smoothness and convexity
properties make the logistic loss preferable to other widely considered options
such as the hinge loss. We show that in contrast to known asymptotic bounds, as
long as the number of prediction/optimization iterations is sub exponential,
the logistic loss provides no improvement over a generic non-smooth loss
function such as the hinge loss. In particular we show that the convergence
rate of stochastic logistic optimization is bounded from below by a polynomial
in the diameter of the decision set and the number of prediction iterations,
and provide a matching tight upper bound. This resolves the COLT open problem
of McMahan and Streeter (2012).
