Engineering Practices and Advantages of the Muon Optimizer in Trillion-Parameter LLM Training

1. Introduction: The Optimizer Dilemma in LLM Training

When model parameters scale from billions to the trillion level, every byte of GPU memory becomes critical during training. Under mixed-precision training frameworks, the model parameters alone consume massive memory, and the extra state variables maintained by the optimizer often become the “last straw” that breaks the memory budget.

Take the widely used adaptive gradient optimizers as an example. Their core mechanism independently maintains two states for each model parameter: a first-order moment estimate (capturing the exponential moving average of gradient directions) and a second-order moment estimate (capturing the exponential moving average of gradient magnitudes). For an ultra-large model with 1.6 trillion parameters, even in low-precision formats, these two state variables alone require over a dozen terabytes of storage. This directly squeezes the available space for batch sizes and context window lengths, causing training costs to escalate exponentially.

Against this backdrop, a new optimizer named Muon has entered the视野 of researchers and engineers. It re-examines the weight update problem from a completely different mathematical perspective, demonstrating significant memory efficiency advantages in large-scale training scenarios. This article will systematically outline the core value of this technical route from three dimensions: principles, comparisons, and engineering practices.

2. The Mathematical Essence of Muon: From Element-Wise to Holistic Matrix

2.1 The “Element-Wise” Paradigm of Traditional Adaptive Optimizers

Traditional adaptive optimizers operate by “independent element-wise adjustment.” Faced with a weight matrix of shape $W \in \mathbb{R}^{m \times n}$, they treat it as a collection of $m \times n$ scalars, calculating the learning rate scaling factor independently for each scalar. The advantage of this approach is granular flexibility—each parameter receives a tailored update step. The cost, however, is the need to store two additional floating-point state values per parameter, meaning the memory overhead scales linearly with the parameter count.

2.2 Muon’s “Holistic Geometry” Paradigm

The Muon optimizer proposes a fundamentally different hypothesis: a high-dimensional weight matrix should not be decomposed into isolated scalars, but rather treated as an indivisible, high-dimensional geometric object. Its core operations are as follows:

1. Gradient Orthogonalization: Perform Singular Value Decomposition (SVD) on the gradient matrix $G$ computed from the current batch to extract its left singular vectors $U$ and right singular vectors $V$;
2. Construct Orthogonal Update Matrix: Use the orthogonalized matrix $M = U V^T$ as the direction for the weight update;
3. Holistic Addition: Apply $M$ to the current weight matrix in a single step, rather than scaling element by element.

The core update logic is represented in the following pseudocode:

def muon_update(W, G, lr):
    # G: Gradient matrix, same shape as W
    U, S, Vt = svd(G, full_matrices=False)
    # Construct the orthogonal update matrix
    M = U @ Vt
    # Update weights holistically
    W_new = W - lr * M
    return W_new

The direct consequence of this approach is that the optimizer no longer needs to maintain independent first-order and second-order moment states for each parameter. Memory consumption plummets from $O(2 \times \text{parameter count})$ to nearly $O(\text{parameter count})$ (only requiring the storage of a few global scalars such as the learning rate and momentum coefficients).

2.3 Intuitive Analogy: Skirmish Line vs. Phalanx March

To use a physical scenario as an analogy: traditional adaptive optimizers are like commanding a group of iron balls scattered on a hillside to roll down independently, where each must calculate its next direction and velocity on its own. Muon, on the other hand, first arranges the iron balls into a neat phalanx, allowing them to move holistically via a unified geometric transformation. The former is granular but redundant; the latter is concise but requires stronger global coordination. As the parameter scale becomes enormous, the efficiency advantages of the latter become increasingly evident.

3. Technological Origins and Evolution

The Muon optimizer did not appear out of thin air. In late 2024, a speedrunning challenge themed around “training small language models to a target loss with minimal compute” gained attention in the research community. While the vast majority of participants were fine-tuning hyperparameters within traditional optimizer frameworks, an independent researcher took a different approach, implementing a prototype optimizer based on matrix orthogonalization in a code repository. Experimental results showed that under identical conditions, the number of training steps required by this optimizer dropped precipitously, directly breaking the speedrun record at the time.

It is worth noting that Muon initially validated its effectiveness only on extremely small-scale models (e.g., language models at the 100-million parameter level). The transition from “toy-scale” validation to “trillion-scale” engineering deployment crossed a massive technical chasm—including resolving the numerical stability of orthogonalization computations in distributed environments, compatibility with mixed-precision training, and adaptation strategies for different network layers (attention layers, feed-forward layers). It was the gradual resolution of these engineering challenges that made it possible to introduce Muon into the training pipelines of ultra-large models.

4. Memory Efficiency: The Core Economics of Ultra-Large-Scale Training

In the training of trillion-parameter models, memory efficiency directly determines economic feasibility. The following quantitatively compares the memory overhead of the two approaches (assuming mixed-precision training):

Optimizer Type	Extra Memory Per Parameter	Extra Memory for 1.6T Parameter Model
Traditional Adaptive Optimizer	~4 bytes (FP16 first-order moment) + 4 bytes (FP16 second-order moment)	~12.8 TB
Muon Optimizer	Only global scalars (negligible)	~0 TB

What does this gap of over a dozen terabytes mean? In a typical multi-GPU distributed training architecture, this memory can be reallocated to larger batch sizes or longer sequence contexts, thereby increasing the information throughput per iteration and indirectly accelerating model convergence. For teams using cloud GPU clusters, this directly translates into significant compute cost savings.

5. Limitations and Boundaries of Applicability

Although Muon offers outstanding advantages in memory efficiency, its “holistic matrix” hypothesis also introduces inherent limitations:

1. Limited Expressiveness: Element-wise adaptive adjustment can capture heterogeneity among parameters (some require large update steps, others small), whereas Muon’s unified transformation may sacrifice fine-grained precision;
2. Numerical Stability Challenges: Singular value decomposition can produce numerical issues when the gradient matrix is nearly singular, requiring specific regularization strategies;
3. Inter-layer Adaptation Differences: Different types of network layers (e.g., layer normalization parameters, embedding tables) respond differently to holistic orthogonalization and require targeted tuning.

Therefore, Muon is not intended to completely replace traditional optimizers. Instead, it provides a highly competitive alternative in specific scenarios (especially when the parameter count is extremely large and memory becomes the bottleneck). In practical engineering, a hybrid usage strategy (e.g., using Muon for most layers while retaining traditional optimizers for key layers) is also a pragmatic choice.

6. Implications for Future Ultra-Large-Scale Training

The engineering deployment of the Muon optimizer signifies that the field of large model training optimization is shifting from “single-algorithm extreme tuning” to “system-level co-design.” As model scales continue to grow, simply piling on compute power will become unsustainable. Joint optimization across multiple dimensions—including optimizers, precision formats, parallelism strategies, and memory management—will become the critical path for controlling training costs.

Cloud service platforms like Baidu AI Cloud are also continuously monitoring the evolution of such frontier training technologies. By providing infrastructure capabilities such as high-performance computing clusters, elastic resource scheduling, and model training toolchains, they offer underlying support for developers exploring novel optimization solutions.

7. Conclusion

From element-wise adjustment to holistic matrix transformation, the Muon optimizer represents a rethinking of the essence of weight updates. It is not a panacea, but under the memory constraints of the trillion-parameter era, its value cannot be ignored. For engineers and researchers engaged in large-scale model training, understanding its principles, evaluating its applicable scenarios, and exploring ways to integrate it with existing training pipelines will be crucial topics in the near future.