Anytime Hedge achieves optimal regret in the stochastic regime

This paper is about a surprising fact: we prove that the anytime Hedge algorithm with decreasing learning rate, which is one of the simplest algorithm for the problem of prediction with expert advice, is actually both worst-case optimal and adaptive to the easier stochastic and adversarial with a gap problems. This runs counter to the common belief in the literature that this algorithm is overly conservative, and that only new adaptive algorithms can simultaneously achieve minimax regret and adapt to the difficulty of the problem. Moreover, our analysis exhibits qualitative differences with other variants of the Hedge algorithm, based on the so-called "doubling trick", and the fixed-horizon version (with constant learning rate).