Article
Version 3
Preserved in Portico This version is not peer-reviewed
From Optimal Control to Optimal Transport via Stochastic Neural Networks in the Mean Field Setting
Version 1
: Received: 2 May 2023 / Approved: 4 May 2023 / Online: 4 May 2023 (10:04:19 CEST)
Version 2 : Received: 6 May 2023 / Approved: 9 May 2023 / Online: 9 May 2023 (04:33:48 CEST)
Version 3 : Received: 20 June 2023 / Approved: 20 June 2023 / Online: 20 June 2023 (11:12:07 CEST)
Version 2 : Received: 6 May 2023 / Approved: 9 May 2023 / Online: 9 May 2023 (04:33:48 CEST)
Version 3 : Received: 20 June 2023 / Approved: 20 June 2023 / Online: 20 June 2023 (11:12:07 CEST)
A peer-reviewed article of this Preprint also exists.
Di Persio, L.; Garbelli, M. From Optimal Control to Mean Field Optimal Transport via Stochastic Neural Networks. Symmetry 2023, 15, 1724. Di Persio, L.; Garbelli, M. From Optimal Control to Mean Field Optimal Transport via Stochastic Neural Networks. Symmetry 2023, 15, 1724.
Abstract
In this paper we derive a unified perspective for Optimal Transport (OT) theory and Mean Field Control (MFC) theory to analyse the learning process for Neural Networks algorithms in a high-dimensional framework. We consider Mean Field Neural Networks in the context of MFC theory, specifically the mean field formulation of OT theory that allows the development of highly efficient algorithms while providing a powerful tool in the context of explainable Artificial Intelligence.
Keywords
Neural Network; Machine Learning; Optimal Transport; Mean Field Control
Subject
Computer Science and Mathematics, Artificial Intelligence and Machine Learning
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Commenter: Matteo Garbelli
Commenter's Conflict of Interests: Author
In particular, we dismissed to consider the causal structure of the learning problem, thus the
Adapted Optimal Transport method, preferring to present a framework able to provide a concrete starting point to foster the inclusion of a temporal structure of marginals within the Optimal Transport setting. We point out that we focused on a measure theory-based formulation for Neural Networks trying to recast the learning problem (and the associated Stochastic Optimal Control) within the Mean Field Optimal Transport theory.