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Multiscale Deep Equilibrium Models


This repo is deprecated and we will soon stop actively maintaining it, as a more up-to-date (and simpler & more efficient) implementation of MDEQ with the same set of tasks as here is now available in the DEQ repo.

We STRONGLY recommend using with the MDEQ-Vision code in the DEQ repo (which also supports Jacobian-related analysis).


This repository contains the code for the multiscale deep equilibrium (MDEQ) model proposed in the paper Multiscale Deep Equilibrium Models by Shaojie Bai, Vladlen Koltun and J. Zico Kolter.

Is implicit deep learning relevant for general, large-scale pattern recognition tasks? We propose the multiscale deep equilibrium (MDEQ) model, which expands upon the DEQ formulation substantially to introduce simultaneous equilibrium modeling of multiple signal resolutions. Specifically, MDEQ solves for and backpropagates through synchronized equilibria of multiple feature representation streams. Such structure rectifies one of the major drawbacks of DEQ, and provide natural hierarchical interfaces for auxiliary losses and compound training procedures (e.g., pretraining and finetuning). Our experiment demonstrate for the first time that "shallow" implicit models can scale to and achieve near-SOTA results on practical computer vision tasks (e.g., megapixel images on Cityscapes segmentation).

We provide in this repo the implementation and the links to the pretrained classification & segmentation MDEQ models.

If you find thie repository useful for your research, please consider citing our work:

    author    = {Shaojie Bai and Vladlen Koltun and J. Zico Kolter},
    title     = {Multiscale Deep Equilibrium Models},
    booktitle   = {Advances in Neural Information Processing Systems (NeurIPS)},
    year      = {2020},


The structure of a multiscale deep equilibrium model (MDEQ) is shown below. All components of the model are shown in this figure (in practice, we use n=4).


Some examples of MDEQ segmentation results on the Cityscapes dataset.


PyTorch >=1.4.0, torchvision >= 0.4.0


  • CIFAR-10: We download the CIFAR-10 dataset using PyTorch's torchvision package (included in this repo).
  • ImageNet We follow the implementation from the PyTorch ImageNet Training repo.
  • Cityscapes: We download the Cityscapes dataset from its official website and process it according to this repo. Cityscapes dataset additionally require a list folder that aligns each original image with its corresponding labeled segmented image. This list folder can be downloaded here.

All datasets should be downloaded, processed and put in the respective data/[DATASET_NAME] directory. The data/ directory should look like the following:

  ...          (other datasets)
  list/        (see above)


All experiment settings are provided in the .yaml files under the experiments/ folder.

To train an MDEQ classification model on ImageNet/CIFAR-10, do

python tools/ --cfg experiments/[DATASET_NAME]/[CONFIG_FILE_NAME].yaml

To train an MDEQ segmentation model on Cityscapes, do

python -m torch.distributed.launch --nproc_per_node=4 tools/ --cfg experiments/[DATASET_NAME]/[CONFIG_FILE_NAME].yaml

where you should provide the pretrained ImageNet model path in the corresponding configuration (.yaml) file. We provide a sample pretrained model extractor in pretrained_models/, but you can also write your own script.

Similarly, to test the model and generate segmentation results on Cityscapes, do

python tools/ --cfg experiments/[DATASET_NAME]/[CONFIG_FILE_NAME].yaml

You can (and probably should) initiate the Cityscapes training with an ImageNet-pretrained MDEQ. You need to extract the state dict from the ImageNet checkpointed model, and set the MODEL.PRETRAINED entry in Cityscapes yaml file to this state dict on disk.

The model implementation and MDEQ's algorithmic components (e.g., L-Broyden's method) can be found in lib/.

Pre-trained Models

We provide some reasonably good pre-trained weights here so that one can quickly play with DEQs without training from scratch.

Description Task Dataset Model
MDEQ-XL ImageNet Classification ImageNet download (.pkl)
MDEQ-XL Cityscapes(val) Segmentation Cityscapes download (.pkl)
MDEQ-Small ImageNet Classification ImageNet download (.pkl)
MDEQ-Small Cityscapes(val) Segmentation Cityscapes download (.pkl)

I. Example of how to evaluate the pretrained ImageNet model:

  1. Download the pretrained ImageNet .pkl file. (I recommend using the gdown command!)
  2. Put the model under pretrained_models/ folder with some file name [FILENAME].
  3. Run the MDEQ classification validation command:
python tools/ --testModel pretrained_models/[FILENAME] --cfg experiments/imagenet/cls_mdeq_[SIZE].yaml

For example, for MDEQ-Small, you should get >75% top-1 accuracy.

II. Example of how to use the pretrained ImageNet model to train on Cityscapes:

  1. Download the pretrained ImageNet .pkl file.
  2. Put the model under pretrained_models/ folder with some file name [FILENAME].
  3. In the corresponding experiments/cityscapes/seg_MDEQ_[SIZE].yaml (where SIZE is typically SMALL, LARGE or XL), set MODEL.PRETRAINED to "pretrained_models/[FILENAME]".
  4. Run the MDEQ segmentation training command (see the "Usage" section above):
python -m torch.distributed.launch --nproc_per_node=[N_GPUS] tools/ --cfg experiments/cityscapes/seg_MDEQ_[SIZE].yaml

III. Example of how to use the pretrained Cityscapes model for inference:

  1. Download the pretrained Cityscapes .pkl file
  2. Put the model under pretrained_models/ folder with some file name [FILENAME].
  3. In the corresponding experiments/cityscapes/seg_MDEQ_[SIZE].yaml (where SIZE is typically SMALL, LARGE or XL), set TEST.MODEL_FILE to "pretrained_models/[FILENAME]".
  4. Run the MDEQ segmentation testing command (see the "Usage" section above):
python tools/ --cfg experiments/cityscapes/seg_MDEQ_[SIZE].yaml


  • To load the Cityscapes pretrained model, download the .pkl file and specify the path in config.[TRAIN/TEST].MODEL_FILE (which is '' by default) in the .yaml files. This is different from setting MODEL.PRETRAINED, see the point below.
  • The difference between [TRAIN/TEST].MODEL_FILE and MODEL.PRETRAINED arguments in the yaml files: the former is used to load all of the model parameters; the latter is for compound training (e.g., when transferring from ImageNet to Cityscapes, we want to discard the final classifier FC layers).
  • The repo supports checkpointing of models at each epoch. One can resume from a previously saved checkpoint by turning on the TRAIN.RESUME argument in the yaml files.
  • Just like DEQs, the MDEQ models can be slower than explicit deep networks, and even more so as the image size increases (because larger images typically require more Broyden iterations to converge well; see Figure 5 in the paper). But one can play with the forward and backward thresholds to adjust the runtime.


Some utilization code (e.g., model summary and yaml processing) of this repo were modified from the HRNet repo and the DEQ repo.


[NeurIPS'20] Multiscale Deep Equilibrium Models







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