Overcoming the Paradox of Certified Training with Gaussian Smoothing

Training neural networks with high certified accuracy against adversarial examples remains an open problem despite significant efforts. While certification methods can effectively leverage tight convex relaxations for bound computation, in training, these methods perform worse than looser relaxations. Prior work hypothesized that this is caused by the discontinuity and perturbation sensitivity of the loss surface induced by these tighter relaxations. In this work, we show theoretically that Gaussian Loss Smoothing can alleviate both issues. We confirm this empirically by proposing a certified training method combining PGPE, an algorithm computing gradients of a smoothed loss, with different convex relaxations. When using this training method, we observe that tighter bounds indeed lead to strictly better networks. While scaling PGPE training remains challenging due to high computational cost, we show that by using a not theoretically sound, yet much cheaper smoothing approximation, we obtain better certified accuracies than state-of-the-art methods when training on the same network architecture. Our results clearly demonstrate the promise of Gaussian Loss Smoothing for training certifiably robust neural networks.