# Generalized Gradient Norm Clipping & Non-Euclidean (L0,L1)-Smoothness

arXiv: https://arxiv.org/abs/2506.01913 · HTML: https://mlsanity.com/paper/2506.01913/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 51 references that are in the map.
- First posted on arXiv 2 Jun 2025.

## Builds on (most cited first, 30 of 32)

- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [Large Batch Training of Convolutional Networks](https://mlsanity.com/paper/1708.03888/) (2017)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [PowerSGD: Practical Low-Rank Gradient Compression for Distributed Optimization](https://mlsanity.com/paper/1905.13727/) (2019)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Momentum Improves Normalized SGD](https://mlsanity.com/paper/2002.03305/) (2020)
- [The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm](https://mlsanity.com/paper/2505.16932/) (2025)
- [Understanding Gradient Orthogonalization for Deep Learning via Non-Euclidean Trust-Region Optimization](https://mlsanity.com/paper/2503.12645/) (2025)
- [Improved Analysis of Clipping Algorithms for Non-convex Optimization](https://mlsanity.com/paper/2010.02519/) (2020)
- [Revisiting Gradient Clipping: Stochastic bias and tight convergence guarantees](https://mlsanity.com/paper/2305.01588/) (2023)
- [A Note on the Convergence of Muon](https://mlsanity.com/paper/2502.02900/) (2025)
- [Robustness to Unbounded Smoothness of Generalized SignSGD](https://mlsanity.com/paper/2208.11195/) (2022)
- [Why Do We Need Weight Decay in Modern Deep Learning?](https://mlsanity.com/paper/2310.04415/) (2023)
- [Scalable Optimization in the Modular Norm](https://mlsanity.com/paper/2405.14813/) (2024)
- [Dion: Distributed Orthonormalized Updates](https://mlsanity.com/paper/2504.05295/) (2025)
- [Implicit Bias of AdamW: ℓ_∞ Norm Constrained Optimization](https://mlsanity.com/paper/2404.04454/) (2024)
- [Convergence of AdaGrad for Non-convex Objectives: Simple Proofs and Relaxed Assumptions](https://mlsanity.com/paper/2305.18471/) (2023)
- [Lion Secretly Solves Constrained Optimization: As Lyapunov Predicts](https://mlsanity.com/paper/2310.05898/) (2023)
- [Stochastic Conditional Gradient Methods: From Convex Minimization to Submodular Maximization](https://mlsanity.com/paper/1804.09554/) (2018)
- [The Geometry of Sign Gradient Descent](https://mlsanity.com/paper/2002.08056/) (2020)
- [Beyond Uniform Smoothness: A Stopped Analysis of Adaptive SGD](https://mlsanity.com/paper/2302.06570/) (2023)
- [Parameter-Agnostic Optimization under Relaxed Smoothness](https://mlsanity.com/paper/2311.03252/) (2023)
- [Orthogonalising gradients to speed up neural network optimisation](https://mlsanity.com/paper/2202.07052/) (2022)
- [On the Overlooked Pitfalls of Weight Decay and How to Mitigate Them: A Gradient-Norm Perspective](https://mlsanity.com/paper/2011.11152/) (2020)
- [Methods for Convex (L0,L1)-Smooth Optimization: Clipping, Acceleration, and Adaptivity](https://mlsanity.com/paper/2409.14989/) (2024)
- [Normalized Gradients for All](https://mlsanity.com/paper/2308.05621/) (2023)
- [An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations](https://mlsanity.com/paper/1304.2338/) (2013)
- [Optimizing (L0, L1)-Smooth Functions by Gradient Methods](https://mlsanity.com/paper/2410.10800/) (2024)
- [Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under (L0, L1)-Smoothness](https://mlsanity.com/paper/2502.07923/) (2025)

## Built on by (most cited first, 17 of 17)

- [Beyond the Ideal: Analyzing the Inexact Muon Update](https://mlsanity.com/paper/2510.19933/) (2025)
- [LiMuon: Light and Fast Muon Optimizer for Large Models](https://mlsanity.com/paper/2509.14562/) (2025)
- [Improved Convergence Rates of Muon Optimizer for Nonconvex Optimization](https://mlsanity.com/paper/2601.19400/) (2026)
- [Optimal Scaling Needs Optimal Norm](https://mlsanity.com/paper/2510.03871/) (2025)
- [Robust Layerwise Scaling Rules by Proper Weight Decay Tuning](https://mlsanity.com/paper/2510.15262/) (2025)
- [On the Role of Batch Size in Stochastic Conditional Gradient Methods](https://mlsanity.com/paper/2603.21191/) (2026)
- [Ky Fan Norms and Beyond: Dual Norms and Combinations for Matrix Optimization](https://mlsanity.com/paper/2512.09678/) (2025)
- [Optimistic Dual Averaging Unifies Modern Optimizers](https://mlsanity.com/paper/2605.11172/) (2026)
- [MiMuon: Mixed Muon Optimizer with Improved Generalization for Large Models](https://mlsanity.com/paper/2605.19619/) (2026)
- [Muon Converges under Heavy-Tailed Noise: Nonconvex H"older-Smooth Empirical Risk Minimization](https://mlsanity.com/paper/2603.15059/) (2026)
- [Gradient Clipping Beyond Vector Norms: A Spectral Approach for Matrix-Valued Parameters](https://mlsanity.com/paper/2605.11838/) (2026)
- [Ringmaster LMO: Asynchronous Linear Minimization Oracle Momentum Method](https://mlsanity.com/paper/2605.18174/) (2026)
- [Free Heavy-Tailed Lunch for Muon: A Theoretical Justification of Empirical Success](https://mlsanity.com/paper/2606.14560/) (2026)
- [Adaptive Conditional Gradient Descent](https://mlsanity.com/paper/2510.11440/) (2025)
- [Distance-Aware Muon: Adaptive Step Scaling for Normalized Optimization](https://mlsanity.com/paper/2605.18999/) (2026)
- [Constrained Stochastic Spectral Preconditioning Converges for Nonconvex Objectives](https://mlsanity.com/paper/2605.11850/) (2026)
- [Can Entry-Wise Clipping Give Spectral Control of Stochastic Gradients?](https://mlsanity.com/paper/2605.27733/) (2026)

## Abstract (arXiv)

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of (L0,L1)-smoothness. Weight decay is incorporated in a principled manner by identifying a connection to the Frank-Wolfe short step. In the stochastic case, we show an order optimal O(n-1/4) convergence rate by leveraging a momentum based gradient estimator. We discuss how to instantiate the algorithms for deep learning, which we dub Clipped Scion, and demonstrate their properties on image classification and language modeling. The code is available at https://github.com/LIONS-EPFL/ClippedScion.