Theoretical Analysis of Robust Overfitting for Wide DNNs: An NTK Approach

Adversarial training (AT) is a canonical method for enhancing the robustness of deep neural networks (DNNs). However, recent studies empirically demonstrated that it suffers from robust <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">overfitting, i.e.</i>, a long time AT can be detrimental to the robustness of DNNs. This paper presents a theoretical explanation of robust overfitting for DNNs. Specifically, we non-trivially extend the neural tangent kernel (NTK) theory to AT and prove that an adversarially trained <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">wide</i> DNN can be well approximated by a linearized DNN. Moreover, for squared loss, closed-form AT dynamics for the linearized DNN can be derived, which reveals a new <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AT degeneration</i> phenomenon: a long-term AT will result in a wide DNN degenerates to that obtained without AT and thus cause robust overfitting. Based on our theoretical results, we further design a method namely <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Adv-NTK</i>, the first AT algorithm for infinite-width DNNs. Experiments on real-world datasets show that Adv-NTK can help infinite-width DNNs enhance comparable robustness to that of their finite-width counterparts, which in turn justifies our theoretical findings. The code is available at https://github.com/fshp971/adv-ntk.