AlphaTensor

This is code accompanying the publication

Fawzi, A. et al. Discovering faster matrix multiplication algorithms with reinforcement learning. Nature 610 (2022)

There are 4 independent directories:

• algorithms contains algorithms discovered by AlphaTensor, represented as factorizations of matrix multiplication tensors, and a Colab showing how to load these.

• benchmarking contains a script that can be used to measure the actual speed of matrix multiplication algorithms on an NVIDIA V100 GPU.

• nonequivalence contains 14,236 nonequivalent algorithms discovered by AlphaTensor for the same matrix multiplication problem (multiplying 4x4 matrices), and a Colab that verifies their nonequivalence.

• recombination contains the code we used to decompose larger matrix multiplication tensors by recombining factorizations of smaller ones.

Installation

• algorithms: No installation required.

• benchmarking: See README in the subdirectory.

• nonequivalence: No installation required.

• recombination: A machine with Python 3 installed is required. The required dependencies (numpy and absl-py) can be installed by executing pip3 install -r alphatensor/recombination/requirements.txt.

Usage

• algorithms: The notebook explore_factorizations.ipynb can be opened via . When running the code, you will be asked to upload a file containing the factorizations. Please select either of the compressed NumPy files factorizations_r.npz (containing algoritms in standard arithmetic) or factorizations_f2.npz (algorithms in arithmetic modulo 2).

• benchmarking: See README in the subdirectory, and Supplement D of the paper.

• nonequivalence: The notebook inspect_factorizations_notebook.ipynb can be opened via . When running the code, you will be asked to upload a file. Please select the compressed NumPy file alphatensor_14236_factorizations.npz. This will upload the factorizations found by AlphaTensor, and then compute invariants certifying that they are all nonequivalent. For more details, see Supplement B of the paper.

• recombination: Execute python3 -m alphatensor.recombination.example on the command line, from the parent directory that contains the alphatensor repository as a subdirectory. For more details, see Supplement H of the paper.

Citing this work

If you use the code or data in this package, please cite:

@Article{AlphaTensor2022,
  author  = {Fawzi, Alhussein and Balog, Matej and Huang, Aja and Hubert, Thomas and Romera-Paredes, Bernardino and Barekatain, Mohammadamin and Novikov, Alexander and Ruiz, Francisco J. R. and Schrittwieser, Julian and Swirszcz, Grzegorz and Silver, David and Hassabis, Demis and Kohli, Pushmeet},
  journal = {Nature},
  title   = {Discovering faster matrix multiplication algorithms with reinforcement learning},
  year    = {2022},
  volume  = {610},
  number  = {7930},
  pages   = {47--53},
  doi     = {10.1038/s41586-022-05172-4}
}

License and disclaimer

Copyright 2022 DeepMind Technologies Limited

All software is licensed under the Apache License, Version 2.0 (Apache 2.0); you may not use this file except in compliance with the Apache 2.0 license. You may obtain a copy of the Apache 2.0 license at: https://www.apache.org/licenses/LICENSE-2.0

All other materials are licensed under the Creative Commons Attribution 4.0 International License (CC-BY). You may obtain a copy of the CC-BY license at: https://creativecommons.org/licenses/by/4.0/legalcode

Unless required by applicable law or agreed to in writing, all software and materials distributed here under the Apache 2.0 or CC-BY licenses are distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the licenses for the specific language governing permissions and limitations under those licenses.

This is not an official Google product.