Regret bounds for Narendra-Shapiro bandit algorithms

Narendra-Shapiro (NS) algorithms are bandit-type algorithms introduced in the sixties (with a view to applications in Psychology or learning automata), whose convergence has been intensively studied in the stochastic algorithm literature. In this talk, we study the efficiency of these bandit algorithms from a regret point of view. We show that some competitive bounds can be obtained for such algorithms in a modified penalized version. Up to an over-penalization modification, the pseudo-regret Rn related to the penalized two-armed bandit is uniformly bounded by C sqrt(n) (for a known C). We also generalize existing convergence and rates of convergence results to the multi-armed case of the over-penalized bandit algorithm, including the convergence toward the invariant measure of a Piecewise Deterministic Markov Process (PDMP) after a suitable renormalization. Finally, ergodic properties of this PDMP are given in the multi-armed case.