TY - JOUR

T1 - A mean-field optimal control formulation of deep learning

AU - Weinan, E.

AU - Han, Jiequn

AU - Li, Qianxiao

N1 - Funding Information:
The work of W. E and J. Han is supported in part by ONR Grant N00014-13-1-0338 and Major Program of NNSFC under Grant 91130005. Q. Li is supported by the Agency for Science, Technology and Research, Singapore.

PY - 2019/3

Y1 - 2019/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85079881749&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85079881749&partnerID=8YFLogxK

U2 - 10.1007/s40687-018-0172-y

DO - 10.1007/s40687-018-0172-y

M3 - Article

AN - SCOPUS:85079881749

VL - 6

JO - Research in Mathematical Sciences

JF - Research in Mathematical Sciences

SN - 2522-0144

IS - 1

M1 - 10

ER -