Neural Network Training as an Optimal Control Problem : — An Augmented Lagrangian Approach —

Training of neural networks amounts to nonconvex optimization problems that are typically solved by using backpropagation and (variants of) stochastic gradient descent. In this work, we propose an alternative approach by viewing the training task as a nonlinear optimal control problem. Under this lens, backpropagation amounts to the sequential approach (single shooting) to optimal control, where the states variables have been eliminated. It is well known that single shooting may lead to ill-conditioning, and for this reason the simultaneous approach (multiple shooting) is typically preferred. Motivated by this hypothesis, an augmented Lagrangian algorithm is developed that only requires an approximate solution to the Lagrangian subproblems up to a user-defined accuracy. By applying this framework to the training of neural networks, it is shown that the inner Lagrangian subproblems are amenable to be solved using Gauss-Newton iterations. To fully exploit the structure of neural networks, the resulting linear least-squares problems are addressed by employing an approach based on forward dynamic programming. Finally, the effectiveness of our method is showcased on regression datasets.