Narendra-Shapiro (NS) algorithms are bandit-type algorithms developed in the 1960s which have been deeply studied in infinite horizon but for which scarce non-asymptotic results exist. In this paper, we focus on a non-asymptotic study of the regret and address the following question: are Narendra-Shapiro bandit algorithms competitive from this point of view? In our main result, we obtain some uniform explicit bounds for the regret of (over)-penalized-NS algorithms. We also extend to the multi-armed case some convergence properties of penalized-NS algorithms towards a stationary Piecewise Deterministic Markov Process (PDMP). Finally, we establish some new sharp mixing bounds for these processes.
Regret; Stochastic Bandit Algorithms; Piecewise Deterministic Markov Processes;
Sébastien Gadat, Fabien Panloup, and Sofiane Saadane, “Regret bound for Narendra-Shapiro bandit algorithms”, TSE Working Paper, n. 15-556, February 2015, revised May 2016.
Fabien Panloup, Sofiane Saadane, and Sébastien Gadat, “Regret bound for Narendra-Shapiro bandit algorithms”, Stochastics, May 2018, pp. 886–926.
Stochastics, May 2018, pp. 886–926