# MAC: An Efficient Gradient Preconditioning using Mean Activation Approximated Curvature

arXiv: https://arxiv.org/abs/2506.08464 · HTML: https://mlsanity.com/paper/2506.08464/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 45 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Builds on (most cited first, 30 of 31)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [mixup: Beyond Empirical Risk Minimization](https://mlsanity.com/paper/1710.09412/) (2017)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [CutMix: Regularization Strategy to Train Strong Classifiers with Localizable Features](https://mlsanity.com/paper/1905.04899/) (2019)
- [RandAugment: Practical automated data augmentation with a reduced search space](https://mlsanity.com/paper/1909.13719/) (2019)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Random Erasing Data Augmentation](https://mlsanity.com/paper/1708.04896/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [TrivialAugment: Tuning-free Yet State-of-the-Art Data Augmentation](https://mlsanity.com/paper/2103.10158/) (2021)
- [ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning](https://mlsanity.com/paper/2006.00719/) (2020)
- [MultiGrain: a unified image embedding for classes and instances](https://mlsanity.com/paper/1902.05509/) (2019)
- [Fast Convergence of Natural Gradient Descent for Overparameterized Neural Networks](https://mlsanity.com/paper/1905.10961/) (2019)
- [Practical Quasi-Newton Methods for Training Deep Neural Networks](https://mlsanity.com/paper/2006.08877/) (2020)
- [M-FAC: Efficient Matrix-Free Approximations of Second-Order Information](https://mlsanity.com/paper/2107.03356/) (2021)
- [Augment your batch: better training with larger batches](https://mlsanity.com/paper/1901.09335/) (2019)
- [Gradient Descent on Neurons and its Link to Approximate Second-Order Optimization](https://mlsanity.com/paper/2201.12250/) (2022)
- [Global Convergence of Adaptive Gradient Methods for An Over-parameterized Neural Network](https://mlsanity.com/paper/1902.07111/) (2019)
- [Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning](https://mlsanity.com/paper/2302.09738/) (2023)

## Built on by (most cited first, 2 of 2)

- [Low-Rank Curvature for Zeroth-Order Optimization in LLM Fine-Tuning](https://mlsanity.com/paper/2511.07971/) (2025)
- [Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluations](https://mlsanity.com/paper/2604.12968/) (2026)

## Abstract (arXiv)

Second-order optimization methods for training neural networks, such as KFAC, exhibit superior convergence by utilizing curvature information of loss landscape. However, it comes at the expense of high computational burden. In this work, we analyze the two components that constitute the layer-wise Fisher information matrix (FIM) used in KFAC: the Kronecker factors related to activations and pre-activation gradients. Based on empirical observations on their eigenspectra, we propose efficient approximations for them, resulting in a computationally efficient optimization method called MAC. To the best of our knowledge, MAC is the first algorithm to apply the Kronecker factorization to the FIM of attention layers used in transformers and explicitly integrate attention scores into the preconditioning. We also study the convergence property of MAC on nonlinear neural networks and provide two conditions under which it converges to global minima. Our extensive evaluations on various network architectures and datasets show that the proposed method outperforms KFAC and other state-of-the-art methods in terms of accuracy, end-to-end training time, and memory usage.