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Adversarial training is well-known to produce high-quality neural networkmodels that are empirically robust against adversarial perturbations.Nevertheless, once a model has been adversarially trained, one often desires acertification that the model is truly robust against all future attacks.Unfortunately, when faced with adversarially trained models, all existingapproaches have significant trouble making certifications that are strongenough to be practically useful. Linear programming (LP) techniques inparticular face a "convex relaxation barrier" that prevent them from makinghigh-quality certifications, even after refinement with mixed-integer linearprogramming (MILP) and branch-and-bound (BnB) techniques. In this paper, wepropose a nonconvex certification technique, based on a low-rank restriction ofa semidefinite programming (SDP) relaxation. The nonconvex relaxation makesstrong certifications comparable to much more expensive SDP methods, whileoptimizing over dramatically fewer variables comparable to much weaker LPmethods. Despite nonconvexity, we show how off-the-shelf local optimizationalgorithms can be used to achieve and to certify global optimality inpolynomial time. Our experiments find that the nonconvex relaxation almostcompletely closes the gap towards exact certification of adversarially trainedmodels.

Tight Certification of Adversarially Trained Neural Networks via Nonconvex Low-Rank Semidefinite Relaxations