Mean-field Langevin System, Optimal Control and Deep Neural Networks
This work addresses the challenge of high-dimensional control optimization, with implications for improving training algorithms in deep learning, though it appears incremental in extending existing mean-field methods to control problems.
The paper tackles the problem of computing optimal controls in high-dimensional settings by introducing a mean-field Langevin system whose invariant measure corresponds to the optimal control, providing a continuous-time algorithm for this purpose. As a result, it validates the solvability of stochastic gradient descent for a broad class of deep neural networks.
In this paper, we study a regularised relaxed optimal control problem and, in particular, we are concerned with the case where the control variable is of large dimension. We introduce a system of mean-field Langevin equations, the invariant measure of which is shown to be the optimal control of the initial problem under mild conditions. Therefore, this system of processes can be viewed as a continuous-time numerical algorithm for computing the optimal control. As an application, this result endorses the solvability of the stochastic gradient descent algorithm for a wide class of deep neural networks.