Understanding Why Adam Outperforms SGD: Gradient Heterogeneity in Transformers

This repository contains the code for our paper:

Understanding Why Adam Outperforms SGD: Gradient Heterogeneity in Transformers. Akiyoshi Tomihari and Issei Sato. arXiv

Dependencies

The main dependencies are:

Python 3.10 or higher
torch = 2.4.0

Please refer to the pyproject.toml file for more details.

Setup

To set up and run the project, follow these steps:

# Configure the project to create virtual environments within the project directory
poetry config virtualenvs.in-project true

# Set the local python version using pyenv
pyenv local 3.12.6

# Install dependencies and activate the virtual environment
poetry install
poetry shell

Training the Models

To train the models, run the following script:

bash shell_scripts/train_<task>.sh <dataset_name> <optimizer_name> <model_name> [<lr_scheduler_type>]

• <task>: Specify nlp or vision.
• <dataset_name>: Name of the dataset (e.g., rte, flowers102).
• <optimizer_name>: Name of the optimizer (e.g., adam, sgd_momentum).
• <model_name>: Name of the model to be trained (e.g., roberta-base, resnet18).
• <lr_scheduler_type> (optional): Learning rate scheduler type, applicable only for NLP tasks. Defaults to default (meaning linear) if not provided.

Example

bash shell_scripts/train_nlp.sh rte adam roberta-base

Calculation of Hessian per Parameter

To calculate the maximum Hessian values for each parameter, run the following script:

bash shell_scripts/hessian_per_param.sh <dataset_name> <optimizer_name> <model_name> <domain> [<training_mode>]

• <domain>: Specify nlp or vision.
• <training_mode> (optional): Specify pretrained to use a pre-trained model. If omitted, a trained model will be used.

When using a trained model, you need to specify the directory in "results/hessian_per_param/model_dir_dict.json".

Example

bash shell_scripts/hessian_per_param.sh rte adam roberta-base pretrained

Acknowledgments

We use the following resources and libraries:

• Base code structure: lp-ft_ntk

• Libraries for NLP tasks: Hugging Face Transformers

• Calculation of Hessian: PyHessian

• Signum optimizer: Signum

Citation

@misc{tomihari2025understandingadamoutperformssgd,
      title={Understanding Why Adam Outperforms SGD: Gradient Heterogeneity in Transformers},
      author={Akiyoshi Tomihari and Issei Sato},
      year={2025},
      eprint={2502.00213},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2502.00213},
}