Recent work linking deep neural networks and dynamical systems opened up new
avenues to analyze deep learning. In particular, it is observed that new
insights can be obtained by recasting deep learning as an optimal control
problem on difference or differential equations. However, the mathematical
aspects of such a formulation have not been systematically explored. This paper
introduces the mathematical formulation of the population risk minimization
problem in deep learning as a mean-field optimal control problem. Mirroring the
development of classical optimal control, we state and prove optimality
conditions of both the Hamilton-Jacobi-Bellman type and the Pontryagin type.
These mean-field results reflect the probabilistic nature of the learning
problem. In addition, by appealing to the mean-field Pontryagin's maximum
principle, we establish some quantitative relationships between population and
empirical learning problems. This serves to establish a mathematical foundation
for investigating the algorithmic and theoretical connections between optimal
control and deep learning.

这篇文章介绍了将深度学习中的群体风险最小化问题作为均场最优控制问题进行数学公式化。通过引用均场 Pontryagin 的最大值原理，确立了种群和经验学习问题之间的一些定量关系，为探究最优控制和深度学习之间的算法和理论联系奠定了数学基础。