FPL Analysis for Adaptive Bandits

A main problem of "Follow the Perturbed Leader" strategies for online decision problems is that regret bounds are typically proven against oblivious adversary. In partial observation cases, it was not clear how to obtain performance guarantees against adaptive adversary, without worsening the bounds. We propose a conceptually simple argument to resolve this problem. Using this, a regret bound of O(t^(2/3)) for FPL in the adversarial multi-armed bandit problem is shown. This bound holds for the common FPL variant using only the observations from designated exploration rounds. Using all observations allows for the stronger bound of O(t^(1/2)), matching the best bound known so far (and essentially the known lower bound) for adversarial bandits. Surprisingly, this variant does not even need explicit exploration, it is self-stabilizing. However the sampling probabilities have to be either externally provided or approximated to sufficient accuracy, using O(t^2 log t) samples in each step.