Adversarially Trained Linear Transformers Are Optimal Robust In-Context Learners for Gaussian Mixtures

Abstract: Adversarial training is one of the most reliable defenses against adversarial attacks, but its high computational cost must generally be paid anew for each task. Robust foundation models offer a promising alternative: adversarially pretrain a model once and then transfer its robustness to downstream tasks through lightweight adaptation. However, a fundamental question remains open: can robustness acquired during pretraining transfer to unseen tasks without further adversarial training? In this study, we answer this question affirmatively. A single model adversarially pretrained at scale can achieve optimal robustness on new tasks without additional task-specific training. Specifically, we show that, for a family of Gaussian-mixture classification tasks, a sufficiently deep linear transformer adversarially trained across tasks can asymptotically attain the robust Bayes error on previously unseen tasks through in-context learning from clean demonstrations. By contrast, a standardly trained model cannot. We further analyze convergence under gradient flow, an accuracy--robustness trade-off, and demonstration complexity.