Fast Certified Robust Training via Better Initialization and Shorter Warmup

Recently, bound propagation based certified adversarial defense have been proposed for training neural networks with certifiable robustness guarantees. Despite state-of-the-art (SOTA) methods including interval bound propagation (IBP) and CROWN-IBP have per-batch training complexity similar to standard neural network training, to reach SOTA performance they usually need a long warmup schedule with hundreds or thousands epochs and are thus still quite costly for training. In this paper, we discover that the weight initialization adopted by prior works, such as Xavier or orthogonal initialization, which was originally designed for standard network training, results in very loose certified bounds at initialization thus a longer warmup schedule must be used. We also find that IBP based training leads to a significant imbalance in ReLU activation states, which can hamper model performance. Based on our findings, we derive a new IBP initialization as well as principled regularizers during the warmup stage to stabilize certified bounds during initialization and warmup stage, which can significantly reduce the warmup schedule and improve the balance of ReLU activation states. Additionally, we find that batch normalization (BN) is a crucial architectural element to build best-performing networks for certified training, because it helps stabilize bound variance and balance ReLU activation states. With our proposed initialization, regularizers and architectural changes combined, we are able to obtain 65.03% verified error on CIFAR-10 ($\epsilon=\frac{8}{255}$) and 82.13% verified error on TinyImageNet ($\epsilon=\frac{1}{255}$) using very short training schedules (160 and 80 total epochs, respectively), outperforming literature SOTA trained with a few hundreds or thousands epochs.