  Recently, bound propagation based certified robust training methods have been
proposed for training neural networks with certifiable robustness guarantees.
Despite that 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, they usually use a long warmup schedule with
hundreds or thousands epochs to reach SOTA performance and are thus still
costly. In this paper, we identify two important issues in existing methods,
namely exploded bounds at initialization, and the imbalance in ReLU activation
states and improve IBP training. These two issues make certified training
difficult and unstable, and thereby long warmup schedules were needed in prior
works. To mitigate these issues and conduct faster certified training with
shorter warmup, we propose three improvements based on IBP training: 1) We
derive a new weight initialization method for IBP training; 2) We propose to
fully add Batch Normalization (BN) to each layer in the model, since we find BN
can reduce the imbalance in ReLU activation states; 3) We also design
regularization to explicitly tighten certified bounds and balance ReLU
activation states during wamrup. We are able to obtain 65.03% verified error on
CIFAR-10 ($\epsilon=\frac{8}{255}$) and 82.36% verified error on TinyImageNet
($\epsilon=\frac{1}{255}$) using very short training schedules (160 and 80
total epochs, respectively), outperforming literature SOTA trained with
hundreds or thousands epochs under the same network architecture. The code is
available at https://github.com/shizhouxing/Fast-Certified-Robust-Training.
