Regret bounds for Narendra-Shapiro bandit algorithms
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Narendra-Shapiro (NS) algorithms are bandit-type algorithms that have been 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 paper, we adress the following question: are the Narendra-Shapiro (NS) bandit algorithms competitive from a regret point of view? In our main result, we show that some competitive bounds can be obtained for such algorithms in their penalized version (introduced in Lamberton_Pages). More precisely, up to an over-penalization modification, the pseudo-regret R̅_n related to the penalized two-armed bandit algorithm is uniformly bounded by C √(n) (where C is made explicit in the paper). 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.
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