When can the two-armed bandit algorithm be trusted?

Details When can the two-armed bandit algorithm be trusted? When can the two-armed bandit algorithm be trusted?

2004-07-08

arxiv.org/abs/math/0407128v1

Annals of Probability 2004, Vol. 14, No. 3, 1424-1454

Mathematics

Probability

Scientific paper

10.1214/105051604000000350

We investigate the asymptotic behavior of one version of the so-called two-armed bandit algorithm. It is an example of stochastic approximation procedure whose associated ODE has both a repulsive and an attractive equilibrium, at which the procedure is noiseless. We show that if the gain parameter is constant or goes to 0 not too fast, the algorithm does fall in the noiseless repulsive equilibrium with positive probability, whereas it always converges to its natural attractive target when the gain parameter goes to zero at some appropriate rates depending on the parameters of the model. We also elucidate the behavior of the constant step algorithm when the step goes to 0. Finally, we highlight the connection between the algorithm and the Polya urn. An application to asset allocation is briefly described.

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