Practical Convex Formulations of One-hidden-layer Neural Network Adversarial Training

As neural networks become more prevalent in safety-critical systems, ensuring their robustness against adversaries becomes essential. "Adversarial training" is one of the most common methods for training robust networks. Current adversarial training algorithms solve highly non-convex bi-level optimization problems. These algorithms suffer from the lack of convergence guarantees and can exhibit unstable behaviors. A recent work has shown that the standard training formulation of a one-hidden-layer, scalar-output fully-connected neural network with rectified linear unit (ReLU) activations can be reformulated as a finite-dimensional convex program, addressing the aforementioned issues for training non-robust networks. In this paper, we leverage this "convex training" framework to tackle the problem of adversarial training. Unfortunately, the scale of the convex training program proposed in the literature grows exponentially in the data size. We prove that a stochastic approximation procedure that scales linearly yields high-quality solutions. With the complexity roadblock removed, we derive convex optimization models that train robust neural networks. Our convex methods provably produce an upper bound on the global optimum of the adversarial training objective and can be applied to binary classification and regression. We demonstrate in experiments that the proposed method achieves a superior robustness compared with the existing methods.