  Follow-the-Leader (FTL) is an intuitive sequential prediction strategy that
guarantees constant regret in the stochastic setting, but has terrible
performance for worst-case data. Other hedging strategies have better
worst-case guarantees but may perform much worse than FTL if the data are not
maximally adversarial. We introduce the FlipFlop algorithm, which is the first
method that provably combines the best of both worlds.
  As part of our construction, we develop AdaHedge, which is a new way of
dynamically tuning the learning rate in Hedge without using the doubling trick.
AdaHedge refines a method by Cesa-Bianchi, Mansour and Stoltz (2007), yielding
slightly improved worst-case guarantees. By interleaving AdaHedge and FTL, the
FlipFlop algorithm achieves regret within a constant factor of the FTL regret,
without sacrificing AdaHedge's worst-case guarantees.
  AdaHedge and FlipFlop do not need to know the range of the losses in advance;
moreover, unlike earlier methods, both have the intuitive property that the
issued weights are invariant under rescaling and translation of the losses. The
losses are also allowed to be negative, in which case they may be interpreted
as gains.
