Weighted Last-Step Min-Max Algorithm with Improved Sub-Logarithmic Regret

TL;DR

This paper introduces a weighted last-step min-max algorithm for online learning that achieves improved logarithmic and sub-logarithmic regret bounds, and extends analysis to non-stationary environments.

Contribution

It fixes a boundedness assumption in Forster's algorithm by weighting examples, providing improved regret bounds and analyzing performance in non-stationary settings.

Findings

Achieves logarithmic regret with better multiplicative factors.

Derives a sub-logarithmic regret bound under certain conditions.

Shows sub-linear regret in non-stationary environments with sub-linear non-stationarity.

Abstract

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.