  The goal of a learner in standard online learning is to maintain an average
loss close to the loss of the best-performing single function in some class. In
many real-world problems, such as rating or ranking items, there is no single
best target function during the runtime of the algorithm, instead the best
(local) target function is drifting over time. We develop a novel last-step
minmax optimal algorithm in context of a drift. We analyze the algorithm in the
worst-case regret framework and show that it maintains an average loss close to
that of the best slowly changing sequence of linear functions, as long as the
total of drift is sublinear. In some situations, our bound improves over
existing bounds, and additionally the algorithm suffers logarithmic regret when
there is no drift. We also build on the H_infinity filter and its bound, and
develop and analyze a second algorithm for drifting setting. Synthetic
simulations demonstrate the advantages of our algorithms in a worst-case
constant drift setting.
