symmetry S S Article Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective Luca Di Persio 1, * and Matteo Garbelli 2   Citation: Di Persio, L.; Garbelli, M. Deep Learning and Mean-Field Games: A Stochastic Optimal Control Perspective. Symmetry 2021, 13, 14. http://dx.doi.org/10.3390/sym 13010014 Received: 18 November 2020 Accepted: 17 December 2020 Published: 23 December 2020 Publisher’s Note: MDPI stays neu- tral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2020 by the authors. Li- censee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). 1 Department of Computer Science, University of Verona, 37134 Verona, Italy; 2 Department of Mathematics, University of Trento, 38123 Trento, Italy; matteo.garbelli@unitn.it * Correspondence: luca.dipersio@univr.it Abstract: 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. Keywords: deep learning; neural networks; stochastic optimal control; mean-field games; Hamilton– Jacobi–Bellman equation; Pontryagin maximum principle 1. Introduction Controlled stochastic processes, which naturally arise in a plethora of heterogeneous fields, spanning, e.g., from mathematical finance to industry, can be solved in the setting of continuous time stochastic control theory. In particular, when we have to analyse complex dynamics produced by the mutual interaction of a large set of indistinguishable players, an efficient approach to infer knowledge about the resulting behaviour, typical for example of a neuronal ensemble, is provided by Mean-Field Game (MFG) methods, as described in [1]. MFG theory generalizes classical models of interacting particle systems character- izing statistical mechanics. Intuitively, each particle is replaced by rational agents whose dynamics are represented by a Stochastic Differential Equation (SDE). The term mean-field refers to the highly symmetric form of interaction: the dynamics and the objective of each particle depend on an empirical measure capturing the global behaviour of the population. The solution of an MFG is analogous to a Nash equilibrium for a non-cooperative game [2]. The key idea is that the population limit can be effectively approximated by statistical features of the system corresponding to the behaviour of a typical group of agents, in a Wasserstein space sense [3]. On the other hand, Deep Learning (DL) is frequently used in several Machine Learning (ML) based applications, spanning from image classification and speech recognition to predictive maintenance and clustering. Therefore, it has become essential to provide a strong mathematical formulation and to analyse both the setting and the associated algorithms [4,5]. Commonly, Neural Networks (NNs) are trained through the Stochastic Gradient Descent (SGD) method. It updates the trainable parameters using gradient information computed randomly via a back-propagation algorithm with the disad- vantage of being slow in the first steps of training. An alternative consists of expressing the learning procedure of an NN as a dynamical system (see [6]), which can be then analysed as an optimal control problem [7]. The present paper is structured as follows. In Section 2, we introduce the fundamentals about the Wasserstein space, Stochastic Optimal Control (SOC) and MFGs. In Section 3, Symmetry 2021, 13, 14. https://dx.doi.org/10.3390/sym13010014 https://www.mdpi.com/journal/symmetry