Positive reinforced generalized time-dependent P\'olya urns via stochastic approximation

TL;DR

This paper extends stochastic approximation methods to analyze a generalized time-dependent Pólya urn process with positive reinforcement, showing that the process eventually fixates on a single urn in a multi-dimensional setting.

Contribution

It introduces a novel extension of stochastic approximation techniques to the d-dimensional case of Pólya urns with positive reinforcement functions.

Findings

The process fixates on a single urn almost surely.

The class of reinforcement functions includes convex and power functions.

The analysis covers time-dependent and generalized reinforcement schemes.

Abstract

Consider a generalized time-dependent P\'olya urn process defined as follows. Let $d\in \mathbb{N}$ be the number of urns/colors. At each time $n$, we distribute $\sigma_n$ balls randomly to the $d$ urns, proportionally to $f$, where $f$ is a valid reinforcement function. We consider a general class of positive reinforcement functions $\mathcal{R}$ assuming some monotonicity and growth condition. The class $\mathcal{R}$ includes convex functions and the classical case $f(x)=x^{\alpha}$, $\alpha>1$. The novelty of the paper lies in extending stochastic approximation techniques to the $d$-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls any more.