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

arXiv: https://arxiv.org/abs/2103.14343 · HTML: https://mlsanity.com/paper/2103.14343/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Mar 2021.

## Built on by (most cited first, 6 of 6)

- [Recurrent Neural Network Training with Convex Loss and Regularization Functions by Extended Kalman Filtering](https://mlsanity.com/paper/2111.02673/) (2021)
- [An Inexact Augmented Lagrangian Algorithm for Training Leaky ReLU Neural Network with Group Sparsity](https://mlsanity.com/paper/2205.05428/) (2022)
- [The inexact power augmented Lagrangian method for constrained nonconvex optimization](https://mlsanity.com/paper/2410.20153/) (2024)
- [Oscillators Are All You Need: Irregular Time Series Modelling via Damped Harmonic Oscillators with Closed-Form Solutions](https://mlsanity.com/paper/2602.12139/) (2026)
- [Augmented Lagrangian Predictive Coding](https://mlsanity.com/paper/2605.31022/) (2026)
- [A proximal augmented Lagrangian method for nonconvex optimization with equality and inequality constraints](https://mlsanity.com/paper/2509.02894/) (2025)

## Abstract (arXiv)

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.