Electronic Journal of Probability

Nonlinear randomized urn models: a stochastic approximation viewpoint

Sophie Laruelle and Gilles Pagès

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This paper extends the link between stochastic approximation ($SA$) theory and randomized urn models developed in [32], and their applications to clinical trials introduced in [2, 3, 4]. We no longer assume that the drawing rule is uniform among the balls of the urn (which contains $d$ colors), but can be reinforced by a function $f$. This is a way to model risk aversion. Firstly, by considering that $f$ is concave or convex and by reformulating the dynamics of the urn composition as an $SA$ algorithm with remainder, we derive the $a.s.$ convergence and the asymptotic normality (Central Limit Theorem, $CLT$) of the normalized procedure by calling upon the so-called $ODE$ and $SDE$ methods. An in depth analysis of the case $d=2$ exhibits two different behaviors: a single equilibrium point when $f$ is concave, and, when $f$ is convex, a transition phase from a single attracting equilibrium to a system with two attracting and one repulsive equilibrium points. The last setting is solved using results on non-convergence toward noisy and noiseless “traps” in order to deduce the $a.s.$ convergence toward one of the attracting points. Secondly, the special case of a Pólya urn (when the addition rule is the $I_{d}$ matrix) is analyzed, still using result from $SA$ theory about “traps”. Finally, these results are used to solve another urn model with a more natural nonlinear drawing rule and we conclude by an example of application to optimal asset allocation in Finance.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 98, 47 pp.

Received: 23 May 2018
Accepted: 29 April 2019
First available in Project Euclid: 18 September 2019

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Zentralblatt MATH identifier

Primary: 62L20: Stochastic approximation 62E20: Asymptotic distribution theory 62L05: Sequential design
Secondary: 62F12: Asymptotic properties of estimators 62P10: Applications to biology and medical sciences

stochastic approximation extended Pólya urn models reinforcement non-homogeneous generating matrix strong consistency asymptotic normality bandit algorithms

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Laruelle, Sophie; Pagès, Gilles. Nonlinear randomized urn models: a stochastic approximation viewpoint. Electron. J. Probab. 24 (2019), paper no. 98, 47 pp. doi:10.1214/19-EJP312. https://projecteuclid.org/euclid.ejp/1568793792

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