Weighted last-step min–max algorithm with improved sub-logarithmic regret

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 [12] proposed a last-step min–max algorithm which was somewhat simpler than the algorithm of Vovk [26], 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 [12] and Vovk [26]. We also derive a new bound that may be sub-logarithmic, as a recent bound of Orabona et al. [21], 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 sublinear if the non-stationarity is sub-linear as well.