Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees

TL;DR

This paper analyzes the convergence of a robust adversarial training method by framing it as an optimal control problem, offering theoretical guarantees and insights into hyperparameter effects, supported by experimental validation.

Contribution

It provides the first convergence analysis of an adversarial training algorithm using optimal control and inexact oracle methods, enhancing understanding of its stability and efficiency.

Findings

Convergence guarantees depend on hyperparameter choices.

Optimal control perspective improves training efficiency.

Experimental results validate theoretical insights.

Abstract

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.