---
title: "Yale Explains Why Muon Beats Adam by Navigating 1.76x Flatter Terrain"
subtitle: "A new paper finally explains why Muon trains LLMs 2x faster than Adam -- the answer lies in how each optimizer navigates the curvature of the loss landscape"
url: "https://alphasignal.ai/news/yale-explains-why-muon-beats-adam-by-navigating-1-76x-flatter-terrain"
author: "AlphaSignal Newsroom"
author_url: "https://alphasignal.ai/editorial-team"
company: "AlphaSignal"
company_url: "https://alphasignal.ai/company/alphasignal"
published: "2026-06-03T00:00:00.000Z"
updated: "2026-06-11T02:44:14.508Z"
type: "Paper"
topics:
  - "Training Infra"
  - "Post Training"
subtopics:
  - "Distributed Training"
  - "Scaling Laws"
  - "Fine Tuning"
upvotes: 2425
source_url: "https://arxiv.org/abs/2606.04662"
comments: 0
truncated: true
---

# Yale Explains Why Muon Beats Adam by Navigating 1.76x Flatter Terrain

> A new paper finally explains why Muon trains LLMs 2x faster than Adam -- the answer lies in how each optimizer navigates the curvature of the loss landscape

Ever since the [Muon optimizer](https://arxiv.org/abs/2502.16982) burst onto the scene and demonstrated roughly a 2x training efficiency gain over Adam for large language models, the ML community has been asking the same question: _why_ does it work so well? The original blog post from Keller Jordan famously attributed Muon's success to "divine benevolence." A new paper from researchers at Yale takes a more rigorous approach, offering the first geometric explanation for Muon's edge -- and the answer comes down to curvature.

## The Problem With "It Just Works"

Despite Muon's empirical success, a rigorous understanding of why and how it outperforms Adam in transformers has remained incomplete. That's a real problem. When you don't understand _why_ an optimizer works, you can't predict when it will fail, how to tune it, or how to improve it. Muon consistently outperforms Adam on various LLM training tasks ranging from small-scale benchmarks to larger settings, and is more data-efficient than Adam, requiring fewer tokens to reach the same validation loss. But the mechanism behind that efficiency has been a black box.

## What Muon Actually Does

To understand the paper's findings, it helps to know what makes Muon different from Adam in the first place. While Adam treats each parameter as an independent scalar, Muon recognizes that neural network weights are matrices with geometric structure -- and optimizes them accordingly.

Concretely, Muon is a matrix orthogonalization-based optimizer specially designed for two-dimensional parameters of hidden layers in a neural network. The key step is orthogonalization: Muon regularizes the gradients with respect to weight matrices by orthogonalization, employing a few Newton-Schulz iterations on the matrix momentum to normalize all singular values to 1\. Think of it as reshaping the gradient update so that no single direction dominates -- every direction of change gets equal weight.

In practice, Muon is typically applied only to 2D weight matrices in hidden layers, while scalar parameters, bias vectors, embeddings, and unembedding layers continue to use standard optimizers such as Adam. So it's not a wholesale replacement -- it's a targeted intervention on the parts of the network where matrix structure matters most.

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## Sources

- [Why Muon Outperforms Adam: A Curvature Perspective](https://arxiv.org/abs/2606.04662)
- [Muon is Scalable for LLM Training](https://arxiv.org/html/2502.16982v1)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://arxiv.org/pdf/2509.26030)
- [Understanding the Muon Optimizer (Medium)](https://jehillparikh.medium.com/understanding-the-muon-optimizer-a-game-changer-for-large-language-model-training-15316975f0bc)
- [Preconditioning Benefits of Spectral Orthogonalization in Muon](https://arxiv.org/pdf/2601.13474)