Sleeping Experts

In “Regret Bounds for Sleeping Experts and Bandits”  Kleinberg, Niculescu-Mizil, and Sharma (2008) propose an algorithm for selecting among a list of experts some of whom may be sleeping.  At each time step $t$, the algorithm ranks the experts in order of preference with a permutation vector $\sigma(t)$.   The first analyzed algorithm is Follow The Awake Leader (FTAL) for the case where all of the expert predictions and rewards are observed.  Next they explore a natural extension of upper confidence bound algorithm for the multi-armed bandit setting.  In the adversarial setting, they prove that the regret is at least $O(\sqrt{t\, K \log K})$ for $K$ bandits at time $t$.