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Optimal learning with non-Gaussian rewards

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  24 March 2016

Zi Ding  and
Ilya O. Ryzhov
Zi Ding*
Affiliation:
University of Maryland
Ilya O. Ryzhov*
Affiliation:
University of Maryland
*
* Postal address: Robert H. Smith School of Business, University of Maryland, 4322 Van Munching Hall, College Park, MD 20742, USA.
* Postal address: Robert H. Smith School of Business, University of Maryland, 4322 Van Munching Hall, College Park, MD 20742, USA.
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Abstract

We propose a novel theoretical characterization of the optimal 'Gittins index' policy in multi-armed bandit problems with non-Gaussian, infinitely divisible reward distributions. We first construct a continuous-time, conditional Lévy process which probabilistically interpolates the sequence of discrete-time rewards. When the rewards are Gaussian, this approach enables an easy connection to the convenient time-change properties of a Brownian motion. Although no such device is available in general for the non-Gaussian case, we use optimal stopping theory to characterize the value of the optimal policy as the solution to a free-boundary partial integro-differential equation (PIDE). We provide the free-boundary PIDE in explicit form under the specific settings of exponential and Poisson rewards. We also prove continuity and monotonicity properties of the Gittins index in these two problems, and discuss how the PIDE can be solved numerically to find the optimal index value of a given belief state.

Keywords

Gittins indicesoptimal learningmulti-armed banditnon-Gaussian rewardsprobabilistic interpolation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J75: Jump processes
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 1 , March 2016 , pp. 112 - 136
Copyright
Copyright © Applied Probability Trust 2016 

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