Tracking the Best Disjunction
Peter L. Auer, Manfred K. Warmuth · Machine Learning · 1998
Littlestone developed a simple deterministic on-line learning algorithm for learning k-literal disjunctions. This algorithm (called $${WINNOW}$$ ) keeps one weight for each of then variables and does multiplicative updates to its weights. We develop a randomized version of $${WINNOW} $$ and prove bounds for an adaptation of the algorithm for the case when the disjunction may change over time. In this case a possible target disjunction schedule $${\mathcal{T}} $$ is a sequence of disjunctions (one per trial) and the shift size is the total number of literals that are added/removed from the disjunctions as one progresses through the sequence. We develop an algorithm that predicts nearly as well as the best disjunction schedule for an arbitrary sequence of examples. This algorithm that allows us to track the predictions of the best disjunction is hardly more complex than the original version. However, the amortized analysis needed for obtaining worst-case mistake bounds requires new techniques. In some cases our lower bounds show that the upper bounds of our algorithm have the right constant in front of the leading term in the mistake bound and almost the right constant in front of the second leading term. Computer experiments support our theoretical findings.