# Gradient Regularized Natural Gradients

arXiv: https://arxiv.org/abs/2601.18420 · HTML: https://mlsanity.com/paper/2601.18420/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 94 references that are in the map.
- First posted on arXiv 26 Jan 2026.

## Builds on (most cited first, 30 of 51)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [DINOv2: Learning Robust Visual Features without Supervision](https://mlsanity.com/paper/2304.07193/) (2023)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Parameter-Efficient Transfer Learning for NLP](https://mlsanity.com/paper/1902.00751/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Parameter-Efficient Fine-Tuning for Large Models: A Comprehensive Survey](https://mlsanity.com/paper/2403.14608/) (2024)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Sophia: A Scalable Stochastic Second-order Optimizer for Language Model Pre-training](https://mlsanity.com/paper/2305.14342/) (2023)
- [On the Origin of Implicit Regularization in Stochastic Gradient Descent](https://mlsanity.com/paper/2101.12176/) (2021)
- [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)
- [Implicit Gradient Regularization](https://mlsanity.com/paper/2009.11162/) (2020)
- [Dissecting Adam: The Sign, Magnitude and Variance of Stochastic Gradients](https://mlsanity.com/paper/1705.07774/) (2017)
- [Why Transformers Need Adam: A Hessian Perspective](https://mlsanity.com/paper/2402.16788/) (2024)
- [Path-SGD: Path-Normalized Optimization in Deep Neural Networks](https://mlsanity.com/paper/1506.02617/) (2015)
- [Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Be](https://mlsanity.com/paper/2304.13960/) (2023)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [The Bayesian Learning Rule](https://mlsanity.com/paper/2107.04562/) (2021)

## Built on by (most cited first, 1 of 1)

- [Guided Riemannian Optimization (GuRO): Bridging Model Predictive Control and Decision Transformers](https://mlsanity.com/paper/2608.23204/) (2026)

## Abstract (arXiv)

Gradient regularization (GR) has been shown to improve the generalizability of trained models. While Natural Gradient Descent has been shown to accelerate optimization in the initial phase of training, little attention has been paid to how the training dynamics of second-order optimizers can benefit from GR. In this work, we propose Gradient-Regularized Natural Gradients (GRNG), a family of scalable second-order optimizers that integrate explicit gradient regularization with natural gradient updates. Our framework introduces two frequentist algorithms: Regularized Explicit Natural Gradient (RENG), which utilizes double backpropagation to explicitly minimize the gradient norm, and Regularized Implicit Natural Gradient (RING), which incorporates regularization implicitly into the update direction. We also propose a Bayesian variant based on a Regularized-Kalman formulation that eliminates the need for FIM inversion entirely. We establish convergence guarantees for GRNG, showing that gradient regularization improves stability and enables convergence to global minima. Empirically, we demonstrate that GRNG consistently enhances both optimization speed and generalization compared to first-order methods (SGD, AdamW) and second-order baselines (K-FAC, Sophia), with strong results on vision and language benchmarks.