# Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees

arXiv: https://arxiv.org/abs/2005.00616 · HTML: https://mlsanity.com/paper/2005.00616/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2020.

## Built on by (most cited first, 7 of 7)

- [Adversarial Training for Gradient Descent: Analysis Through its Continuous-time Approximation](https://mlsanity.com/paper/2105.08037/) (2021)
- [Neural Predictive Control to Coordinate Discrete- and Continuous-Time Models for Time-Series Analysis with Control-Theoretical Improvements](https://mlsanity.com/paper/2508.01833/) (2025)
- [Semi-Implicit Hybrid Gradient Methods with Application to Adversarial Robustness](https://mlsanity.com/paper/2202.10523/) (2022)
- [Extracting Global Dynamics of Loss Landscape in Deep Learning Models](https://mlsanity.com/paper/2106.07683/) (2021)
- [On the Escaping Efficiency of Distributed Adversarial Training Algorithms](https://mlsanity.com/paper/2509.11337/) (2025)
- [Optimal Control Theoretic Neural Optimizer: From Backpropagation to Dynamic Programming](https://mlsanity.com/paper/2510.14168/) (2025)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

The fragility of deep neural networks to adversarially-chosen inputs has motivated the need to revisit deep learning algorithms. Including adversarial examples during training is a popular defense mechanism against adversarial attacks. This mechanism can be formulated as a min-max optimization problem, where the adversary seeks to maximize the loss function using an iterative first-order algorithm while the learner attempts to minimize it. However, finding adversarial examples in this way causes excessive computational overhead during training. By interpreting the min-max problem as an optimal control problem, it has recently been shown that one can exploit the compositional structure of neural networks in the optimization problem to improve the training time significantly. In this paper, we provide the first convergence analysis of this adversarial training algorithm by combining techniques from robust optimal control and inexact oracle methods in optimization. Our analysis sheds light on how the hyperparameters of the algorithm affect the its stability and convergence. We support our insights with experiments on a robust classification problem.