Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective

We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we sh...

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Bibliographic Details
Published in:Symmetry (Basel) 2021-01, Vol.13 (1), p.14
Main Authors: Persio, Luca Di, Garbelli, Matteo
Format: Article
Language:English
Subjects:
deep learning
Hamilton–Jacobi–Bellman equation
mean-field games
neural networks
Pontryagin maximum principle
stochastic optimal control
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Summary:We provide a rigorous mathematical formulation of Deep Learning (DL) methodologies through an in-depth analysis of the learning procedures characterizing Neural Network (NN) models within the theoretical frameworks of Stochastic Optimal Control (SOC) and Mean-Field Games (MFGs). In particular, we show how the supervised learning approach can be translated in terms of a (stochastic) mean-field optimal control problem by applying the Hamilton–Jacobi–Bellman (HJB) approach and the mean-field Pontryagin maximum principle. Our contribution sheds new light on a possible theoretical connection between mean-field problems and DL, melting heterogeneous approaches and reporting the state-of-the-art within such fields to show how the latter different perspectives can be indeed fruitfully unified.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13010014