Neural networks are vulnerable to adversarial examples, i.e. inputs that are imperceptibly perturbed from natural data and yet incorrectly classified by the network. Adversarial training, a heuristic form of robust optimization that alternates between minimization and maximization steps, has proven to be among the most successful methods to train networks that are robust against a pre-defined family of perturbations. This paper provides a partial answer to the success of adversarial training. When the inner maximization problem can be solved to optimality, we prove that adversarial training finds a network of small robust train loss. When the maximization problem is solved by a heuristic algorithm, we prove that adversarial training finds a network of small robust surrogate train loss. The analysis technique leverages recent work on the analysis of neural networks via Neural Tangent Kernel (NTK), combined with online-learning when the maximization is solved by a heuristic, and the expressiveness of the NTK kernel in the ℓ∞-norm.