  In online learning the performance of an algorithm is typically compared to
the performance of a fixed function from some class, with a quantity called
regret. Forster proposed a last-step min-max algorithm which was somewhat
simpler than the algorithm of Vovk, yet with the same regret. In fact the
algorithm he analyzed assumed that the choices of the adversary are bounded,
yielding artificially only the two extreme cases. We fix this problem by
weighing the examples in such a way that the min-max problem will be well
defined, and provide analysis with logarithmic regret that may have better
multiplicative factor than both bounds of Forster and Vovk. We also derive a
new bound that may be sub-logarithmic, as a recent bound of Orabona et.al, but
may have better multiplicative factor. Finally, we analyze the algorithm in a
weak-type of non-stationary setting, and show a bound that is sub-linear if the
non-stationarity is sub-linear as well.
