  Recent empirical and theoretical studies have established the generalization
capabilities of large machine learning models that are trained to
(approximately or exactly) fit noisy data. In this work, we prove a surprising
result that even if the ground truth itself is robust to adversarial examples,
and the benignly overfitted model is benign in terms of the ``standard''
out-of-sample risk objective, this benign overfitting process can be harmful
when out-of-sample data are subject to adversarial manipulation. More
specifically, our main results contain two parts: (i) the min-norm estimator in
overparameterized linear model always leads to adversarial vulnerability in the
``benign overfitting'' setting; (ii) we verify an asymptotic trade-off result
between the standard risk and the ``adversarial'' risk of every ridge
regression estimator, implying that under suitable conditions these two items
cannot both be small at the same time by any single choice of the ridge
regularization parameter. Furthermore, under the lazy training regime, we
demonstrate parallel results on two-layer neural tangent kernel (NTK) model,
which align with empirical observations in deep neural networks. Our finding
provides theoretical insights into the puzzling phenomenon observed in
practice, where the true target function (e.g., human) is robust against
adverasrial attack, while beginly overfitted neural networks lead to models
that are not robust.
