Generic Behaviour of Strongly Reinforced P\'{o}lya Urns : Convergence and Stability

Date(s) - 11/10/2019
01:30 - 15:30

Bilkent-Unv EA409

Categories No Categories

Assoc. Prof. Dr. Arnab Basu

Industrial Engineering, Bilkent University

Oct 11, 1:40 p.m.


We consider a class of graph-based “interacting urn”-type P\'{o}lya urn model inspired by neuronal processing in the brain where a signal enters the brain at some (randomly) chosen neuron and is transmitted to a (random) single neighbouring neuron with a probability depending on the relative `efficiency’ of the synapses connecting the neurons, and in doing so the efficiency of the utilized synapse is improved / reinforced. We study the structures (or architectures) and relative efficiency of the neuronal networks that can arise from repeating this process a very large number of times in a “strong reinforcement regime”. Under the most general conditions, we prove in the affirmative a part of the main open conjecture of the strict positivity of the probability of convergence of the corresponding “urn process” to any given `stable’ equilibrium and the zero probability of convergence of the corresponding “urn process” to any `unstable’ equilibrium. Under very generic conditions, i.e., for an open and dense subset of parameters with full measure, we also prove the full open conjecture by showing the finiteness of the equilibrium set and hence the unit probability of convergence of the “urn process” to some `stable’ equilibrium. We borrow ideas from Dynamical Systems and Differential Topology in this analysis of our problem.

Arnab Basu is currently a Visting Associate Professor (Full-time) in the Industrial Engineering Department at Bilkent University.He has been a permanent (tenured) Associate Professor of Decision Sciences at the Indian Institute of Management Bangalorein India till September 2019. Arnab’s current research focus is in understanding how learning happens in mutli-agent systems under randomness and competition. He has been working in several areas of Stochastic Control and Games and has made deep contributions to the area of Risk-sensitive Stochastic Games by solving the open problems of the existence of equilibrium in such games in a series of publications. He has been and still is a Visiting Research Faculty to several universites in the EU. The applications of his work include, among others, AI / ML in Computer Science, General Equilibrum Models in Economics, and, Risk Estimation and Portfolio Optimization in Finance. Given the interdisciplinary nature of his research, he uses ideas from Topology, Geometry and Dynamical Systems in the modelling and analysis of such problems.