# Beyond the Mean: Fisher-Orthogonal Projection for Natural Gradient Descent in Large Batch Training

arXiv: https://arxiv.org/abs/2508.13898 · HTML: https://mlsanity.com/paper/2508.13898/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 34 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Builds on (most cited first, 24 of 24)

- [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)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Class-Balanced Loss Based on Effective Number of Samples](https://mlsanity.com/paper/1901.05555/) (2019)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [Decoupling Representation and Classifier for Long-Tailed Recognition](https://mlsanity.com/paper/1910.09217/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Large-Scale Long-Tailed Recognition in an Open World](https://mlsanity.com/paper/1904.05160/) (2019)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [Revisiting Natural Gradient for Deep Networks](https://mlsanity.com/paper/1301.3584/) (2013)
- [Eigenvalues of the Hessian in Deep Learning: Singularity and Beyond](https://mlsanity.com/paper/1611.07476/) (2016)
- [MLPerf Training Benchmark](https://mlsanity.com/paper/1910.01500/) (2019)
- [A Distributed Data-Parallel PyTorch Implementation of the Distributed Shampoo Optimizer for Training Neural Networks At-Scale](https://mlsanity.com/paper/2309.06497/) (2023)
- [ASDL: A Unified Interface for Gradient Preconditioning in PyTorch](https://mlsanity.com/paper/2305.04684/) (2023)
- [Large batch size training of neural networks with adversarial training and second-order information](https://mlsanity.com/paper/1810.01021/) (2018)
- [Scalable and Practical Natural Gradient for Large-Scale Deep Learning](https://mlsanity.com/paper/2002.06015/) (2020)
- [Sketchy Empirical Natural Gradient Methods for Deep Learning](https://mlsanity.com/paper/2006.05924/) (2020)
- [Structured Inverse-Free Natural Gradient: Memory-Efficient & Numerically-Stable KFAC](https://mlsanity.com/paper/2312.05705/) (2023)

## Built on by (most cited first, 5 of 5)

- [Fisher-Orthogonal Projected Natural Gradient Descent for Continual Learning](https://mlsanity.com/paper/2601.12816/) (2026)
- [Second-Order Multi-Level Variance Correction for Modality Competition in Multimodal Models](https://mlsanity.com/paper/2605.16165/) (2026)
- [DyKAF: Dynamical Kronecker Approximation of the Fisher Information Matrix for Gradient Preconditioning](https://mlsanity.com/paper/2511.06477/) (2025)
- [Runtime-Orchestrated Second-Order Optimization for Scalable LLM Training](https://mlsanity.com/paper/2605.16184/) (2026)
- [Detecting Contaminated Code-Generation Prompt Batches via Influence Functions](https://mlsanity.com/paper/2608.14303/) (2026)

## Abstract (arXiv)

Modern GPUs are equipped with large amounts of high-bandwidth memory, enabling them to support mini-batch sizes of up to tens of thousands of training samples. However, most existing optimizers struggle to perform effectively at such a large batch size. As batch size increases, gradient noise decreases due to averaging over many samples, limiting the ability of first-order methods to escape sharp or suboptimal minima and reach the global minimum. Meanwhile, second-order methods like the natural gradient with Kronecker-Factored Approximate Curvature (KFAC) often require excessively high damping to remain stable at large batch sizes. This high damping effectively washes out the curvature information that gives these methods their advantage, reducing their performance to that of simple gradient descent. In this paper, we introduce Fisher-Orthogonal Projection (FOP), a novel technique that restores the effectiveness of the second-order method at very large batch sizes, enabling scalable training with improved generalization and faster convergence. FOP constructs a variance-aware update direction by leveraging gradients from two sub-batches, enhancing the average gradient with a component of the gradient difference that is orthogonal to the average under the Fisher-metric.