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CROWN-IBP: Towards Stable and Efficient Training of Verifiably Robust Neural Networks

CROWN-IBP is a certified defense to adversarial examples, which combines a tight linear relaxation based verification bound (CROWN) with the efficient interval bound propagation (IBP) training method. We achieved state-of-the-art verified (certified) error on MNIST and CIFAR: for MNIST, 6.68% at epsilon=0.3 and 12.46% at epsilon=0.4 (L_infinity norm distortion); and for CIFAR, 67.11% at epsilon=8/255 and 45.50% at epsilon=2/255. The MNIST verified error is even lower than the unverified PGD error (around 12% at epsilon=0.3) by (Madry et al.). More empirical results and algorithm details of CROWN-IBP can be found in our paper:

Huan Zhang, Hongge Chen, Chaowei Xiao, Sven Gowal, Robert Stanforth, Bo Li, Duane Boning, and Cho-Jui Hsieh, "Towards Stable and Efficient Training of Verifiably Robust Neural Networks", ICLR 2020

Our repository provides high quality PyTorch implementations of IBP (Gowal et al., 2018), CROWN-IBP (our algorithm), Convex Adversarial Polytope (Wong et al., 2018) and (ordinary) CROWN (Zhang et al., 2018) with Multi-GPU support.

If you want to apply CROWN-IBP to more general NN architectures (e.g., ResNet/DenseNet, LSTM and Transformer), use the new implementation based on the auto_LiRPA library. auto_LiRPA is a powerful tool for computing linear relaxation based perturbation analysis (e.g., CROWN) for general computational graphs.

We recently proposed a new technique, loss fusion, which further improves the efficiency of CROWN-IBP. We can now use the tight CROWN-IBP bounds for training at the same asympototic complexity as IBP, and scale to large datasets including downscaled ImageNet. More details:

A TensorFlow implementation of CROWN-IBP is provided by DeepMind.


  • Apr 24, 2020: A new implementation of CROWN-IBP on more neural network architectures (ResNet, LSTM, Transformer) can be found in the auto_LiRPA library.
  • Nov 21, 2019: We released pretrained SOTA models. On MNIST we can achieve 6.68% verified error at epsilon=0.3 and on CIFAR we can achieve 67.11% verified error at epsilon=8/255. Follow instructions here.
  • Nov 21, 2019: Multi-GPU training support. It is now possible to scale CROWN-IBP to the same Large model (trained on 32 TPUs) reported by Gowal et al., 2018. The largest CIFAR model takes about 1 day to train on 4x 2080 Ti GPUs.
  • Oct 10, 2019: Ordinary CROWN bounds have been added for verification propose. See instructions. The implementation takes advantage of CNN on GPUs and is efficient for verification.
  • July 14, 2019: Code has been further optimized and CROWN-IBP training is roughly 2-3 times faster than before.
  • Jun 8, 2019: Initial release.


We aim to train a robust deep neural network that is verifiable: given a test example and a certain perturbation radius epsilon, it is able to verify if there is definitely no adversarial example inside, or there might exist an adversarial example, through some efficient neural network verification algorithm. Conversely, PGD based adversarial training is generally not a verifiable training method, as existing verification methods typically give vacuous bounds for a PGD-trained network, and there might exists stronger attacks that can break a PGD-trained model.

The certified robustness of a model can be evaluated using verified error, which is a guaranteed upper bound of test error under any attack. CROWN-IBP can achieve 6.68% verified error on MNIST test set at epsilon=0.3. The verified error is even lower than the unverified PGD error (around 12%) provided by Adversarial training using PGD-based adversarial training. Previous convex relaxation based method by Wong et al. has about 30% to 40% verified error.

CROWN-IBP combines Interval Bound Propagation (IBP) training (Gowal et al. 2018) and CROWN (Zhang et al. 2018), a tight linear relaxation based verification method. Ordinary CROWN is very expensive for training, and it may also over-regularize the network. IBP alone outperforms many existing linear relaxation based methods due to its non-linear representation power (see our paper for more explanation), however since IBP bounds are very loose at the beginning phase of training, training stability is a big concern. We propose to use IBP in a forward pass to compute all intermediate layers' pre-activation bounds and use a CROWN-style bound to obtain the final layer's bounds in a backward pass. The combination of IBP and CROWN gives us an efficient, stable and well-performing certified defense method, strictly within the framework of robust optimization.

This repository also includes a Pytorch implementation of (ordinary) CROWN, which computes the full convex relaxation based bounds and can be used to compute CROWN verified error. Although it is also okay to use the ordinary CROWN bounds for training (the code supports it, see example config files in experimental folder), but it is very inefficient comparing to CROWN-IBP, and also produces models with worse verified error due to over-regularization.

Getting Started with the Code

Our program is tested on Pytorch 1.3.0 and Python 3.6/3.7.

We have all training parameters included in JSON files, under the config directory. We provide configuration files which can reproduce all CROWN-IBP models in our paper.

To train CROWN-IBP on MNIST, 10 small models, run:

python --config config/mnist_crown.json

To train CROWN-IBP on MNIST, 8 medium models, run:

python --config config/mnist_crown_large.json

To train CROWN-IBP on MNIST, using the very large model in Gowal et al. to achieve SOTA, run:

# This uses all GPUs by default.
python --config config/mnist_dm-large_0.4.json

To train CROWN-IBP on CIFAR, using the very large model in Gowal et al. to achieve SOTA, run:

# This uses all GPUs by default.
python --config config/cifar_dm-large_8_255.json

You will also find configuration files for 9 small CIFAR models, 8 medium CIFAR models, 10 Fashion-MNIST small models and 8 Fashion-MNIST medium models in config folder. All hyperparameters can be changed in the configuration JSON file.

Pre-trained Models

CROWN-IBP pretrained models (small and medium sized) used in our paper can be downloaded here. The very large model structure (used in Gowal et al., 2018) trained using CROWN-IBP can be downloaded here.

# Large SOTA models
tar xvf models_crown-ibp_dm-large.tar.gz
# Small and medium sized models
tar xvf models_crown-ibp.tar.gz

To evaluate the best (and largest) MNIST and CIFAR model (same model structure as in Gowal et al. 2018, referred to as "dm-large" in our paper), run:

# Evaluate MNIST with epsilon=0.3
# The default epsilon for MNIST evaluation in config/mnist_dm-large_0.4.json is 0.3.
python --config config/mnist_dm-large_0.4.json --path_prefix models_crown-ibp_dm-large
# Evaluate MNIST with epsilon=0.4
python "eval_params:epsilon=0.4" --config config/mnist_dm-large_0.4.json --path_prefix models_crown-ibp_dm-large
# Evaluate CIFAR-10 with epsilon=8/255
python --config config/cifar_dm-large_8_255.json  --path_prefix models_crown-ibp_dm-large
# Evaluate CIFAR-10 with epsilon=2/255
python --config config/cifar_dm-large_2_255.json  --path_prefix models_crown-ibp_dm-large

Note that the "dm-large" models have slight different verified errors than the models reported in our paper (accuracy within +/-0.5%), which were trained using Tensorflow. The default epsilon for evaluation in config/mnist_dm-large_0.4.json is 0.3. The parameter "eval_params:epsilon=0.4" overrides the epsilon in configuration file, and the parameter --path_prefix changes the default path that stores models and logs.

The folder crown-ibp_models contains several directories, each one corresponding to a set of (relatively small) models and a epsilon value. They can also be evaluated using the script For example:

# Evaluate the 8 medium MNIST models under epsilon=0.4
python "eval_params:epsilon=0.4" --config config/mnist_crown_large.json --path_prefix crown-ibp_models/mnist_0.4_mnist_crown_large
# Evaluate the 8 medium CIFAR models under epsilon=0.03137 (8/255)
# No epsilon value is given explicitly, so the default in cifar_crown_large.json will be used
python --config config/cifar_crown_large.json --path_prefix crown-ibp_models/cifar_crown_large_0.03137/

The script will report min, median and max verified errors across all models.

Training options

Training verified models using CROWN-IBP is easy: simply use with a config file, which includes necessary training parameters. For example:

# Train the largest MNIST model ("dm-large")
python --config config/mnist_dm-large_0.4.json
# Train the largest CIFAR-10 model ("dm-large")
python --config config/cifar_dm-large_8_255.json

Multiple models may be defined in a confg file, to train the a specific model, use the --model_subset argument:

python --config config/mnist_crown_large.json --model_subset 4

The argument --model_subset selects the 4th model defined in configuration file. To change perturbation epsilon, use the parameter "training_params:epsilon=0.4" which overrides the corresponding epsilon value in configuration file. Other parameters can be overridden in a similar manner. For epsilon=0.4 run this command:

python "training_params:epsilon=0.4" --config config/mnist_crown_large.json --model_subset 4

If you want to use multiple GPUs for training, set keyword "multi_gpu" under "training_params" section in configuration as true, or equivalently add "training_params:multi_gpu=true" in command line after Then the program uses all available GPUs in the system. To use less than all GPUs, set environment variable CUDA_VISIBLE_DEVICES before you run.

We also implement baseline methods including IBP (Gowal et al. 2018) and Convex adversarial polytope (Wong et al. 2018). They can be used by adding command line parameters to override the training method defined in the JSON file. For example,

# for IBP (no kappa terms)
python "training_params:method_params:bound_type=interval" --config config/mnist_crown.json 
# for IBP (with kappa terms as in Gowal et al., 2018)
python "training_params:method=robust_natural" "training_params:method_params:bound_type=interval" --config config/mnist_crown.json
# for Convex adversarial polytope (Wong et al. 2018)
python "training_params:method_params:bound_type=convex-adv" --config config/mnist_crown.json

Training Your Own Robust Model Using CROWN-IBP

Our implementation can be easily extended to other datasets or models. You only need to do three things:

  • Add your dataset loader to (see examples for MNIST, Fashion-MNIST, CIFAR-10 and SVHN)
  • Add your model architecture to (you can also reuse any existing models)
  • Create a JSON configuration file for training. You can copy from crown_mnist.json or crown_cifar.json. You need to change "dataset" name in config file, also update model structure in the "models" section.

For example, the following in "models" section of "mnist_crown.json" defines a model with 2 CNN layers:

    "model_id": "cnn_2layer_width_1",
    "model_class": "model_cnn_2layer",
    "model_params": {"in_ch": 1, "in_dim": 28, "width": 1, "linear_size": 128}

where "model_id" is an unique name for each model, "model_class" is the function name to create the model in, and "model_params" is a dictionary of all parameters passing to the function that creates the model. Then your will be able to train with CROWN-IBP with your JSON:

python --config your_config.json

Compute CROWN Verified Errors

Unlike the reference implementation in (Zhang et al., 2018) which only supports linear layers and is implemented in Numpy, this repository provides an efficient Pytorch implementation of CROWN for convolutional neural networks.

In the default setting, the code evaluates IBP verified error. To compute CROWN verified error, simply set eval_params:method_params:bound_type=crown-full in the evaluation command. Also, you may need to reduce batch size by setting eval_params:loader_params:test_batch_size to a small value, since the CROWN bounds are memory intensive to compute.


python "eval_params:epsilon=0.3" "eval_params:loader_params:test_batch_size=128" "eval_params:method_params:bound_type=crown-full" --config config/mnist_crown.json --path_prefix crown-ibp_models/mnist_0.3_mnist_crown --model_subset 1

Note that CROWN bounds are relatively loose on (CROWN-)IBP trained models; the above command produces verified error around 50% (while IBP verified error is around 10%). However, CROWN should achieve much tighter bounds on naturally trained, PGD adversarially trained or randomly initialized networks, and is an useful tool for giving linearized upper and lower bounds for general neural networks.

Computing CROWN verified error on the entire test set (10,000 images) can take some time (a few minutes to a few hours). The time is similar to 1 epoch training time of (Wong & Kolter, ICML 2018), so it is still feasible to run all 10,000 examples for most models.

Several bound options for CROWN are provided:

same-slope: when set to true, use the same lower bound slope as the upper bound slope for unstable ReLU neurons. This reduces computation cost by a half, but usually leads to worse verified error.

zero-lb: set the lower bound slope to 0 for unstable ReLU neurons. CROWN provides a valid bound for any slope from 0 to 1, and these slopes correspond to dual variables in dual LP formulation; see (Salman et al. 2019) for more details. Sometimes setting this option to true can improve verified error.

one-lb: set the lower bound slope to 1 for unstable ReLU neurons. Sometimes setting this option to true can improve verified error.

Example (setting zero-lb to true to improve CROWN verified error from 53.31% to 10.61%):

python "eval_params:method_params:bound_opts:zero-lb=true" "eval_params:epsilon=0.3" "eval_params:loader_params:test_batch_size=128" "eval_params:method_params:bound_type=crown-full" --config config/mnist_crown.json --path_prefix crown-ibp_models/mnist_0.3_mnist_crown --model_subset 1

Reproducing Paper Results on Different Kappa and Training Schedules

In our paper, we evaluate training stability by setting different epsilon schedule length among different models and methods. epsilon schedule length can be controlled through changing the configuration file, or overriding configuration file parameters as shown below.

To train CROBW-IBP on MNIST, 10 small models with epsilon=0.3 and schedule length as 10, run this command:

python training_params:schedule_length=11 --config config/mnist_crown.json 

To train IBP on MNIST with no natural CE loss, 10 small models with epsilon=0.3 and schedule length as 10, run this command:

python training_params:schedule_length=11 training_params:method_params:bound_type=interval --config config/mnist_crown.json 

To train IBP with final kappa=0.5 on MNIST, 10 small models with epsilon=0.3 and schedule length as 10, run this command:

python training_params:schedule_length=11 training_params:method_params:bound_type=interval training_params:method_params:final-kappa=0.5 training_params:method=robust_natural --config config/mnist_crown.json  

To train IBP with final kappa=0 on MNIST, 10 small models with epsilon=0.3 and schedule length as 10, run this command:

python training_params:schedule_length=11 training_params:method_params:bound_type=interval training_params:method_params:final-kappa=0 training_params:method=robust_natural --config config/mnist_crown.json


We stand on the shoulders of giants and we greatly appreciate the inspiring works of pioneers in this field. Here we list references directly related to this README. A full bibliography can be found in our paper.

Sven Gowal, Krishnamurthy Dvijotham, Robert Stanforth, Rudy Bunel, Chongli Qin, Jonathan Uesato, Timothy Mann, and Pushmeet Kohli. "On the effectiveness of interval bound propagation for training verifiably robust models." arXiv preprint arXiv:1810.12715 (2018).

Eric Wong, and J. Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. ICML 2018.

Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. "Towards deep learning models resistant to adversarial attacks." In International Conference on Learning Representations, 2018.

Huan Zhang, Tsui-Wei Weng, Pin-Yu Chen, Cho-Jui Hsieh, and Luca Daniel. Efficient neural network robustness certification with general activation functions. In Advances in neural information processing systems (NIPS), pp. 4939-4948. 2018.

Hadi Salman, Greg Yang, Huan Zhang, Cho-Jui Hsieh and Pengchuan Zhang. A Convex Relaxation Barrier to Tight Robustness Verification of Neural Networks. To appear in NeurIPS 2019.


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