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Lizhang Chen

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4 papers

2 author rows

TMLR Journal 2026 Journal Article

Muon Optimizes Under Spectral Norm Constraints

Lizhang Chen
Jonathan Li
Qiang Liu

The pursuit of faster optimization algorithms remains an active and important research direction in deep learning. Recently, the Muon optimizer has demonstrated promising empirical performance, but its theoretical foundation remains less understood. In this paper, we bridge this gap and provide a theoretical analysis of Muon by placing it within the Lion-$\mathcal{K}$ family of optimizers. Specifically, we show that Muon corresponds to Lion-$\mathcal{K}$ when equipped with the nuclear norm, and we leverage the theoretical results of Lion-$\mathcal{K}$ to establish that Muon (with decoupled weight decay) implicitly solves an optimization problem that enforces a constraint on the spectral norm of weight matrices. This perspective not only demystifies the implicit regularization effects of Muon but also leads to natural generalizations through varying the choice of convex map $\mathcal{K}$, allowing for the exploration of a broader class of implicitly regularized and constrained optimization algorithms.

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NeurIPS Conference 2024 Conference Paper

Communication Efficient Distributed Training with Distributed Lion

Bo Liu
Lemeng Wu
Lizhang Chen
Kaizhao Liang
Jiaxu Zhu
Chen Liang
Raghuraman Krishnamoorthi
Qiang Liu

The Lion optimizer has been a promising competitor with the AdamW for training large AI models, with advantages in memory, computation, and sample efficiency. In this paper, we introduce Distributed Lion, an innovative adaptation of Lion for distributed training environments. Leveraging the sign operator in Lion, our Distributed Lion only requires to communicate binary or lower-precision vectorsbetween workers to the center server, significantly reducing the communication cost. Our theoretical analysis confirms Distributed Lion's convergence properties. Empirical results demonstrate its robustness across a range of tasks, worker counts, and batch sizes, on both vision and language problems. Notably, Distributed Lion attains comparable performance to standard Lion or AdamW optimizers applied on aggregated gradients, but with significantly reduced communication bandwidth. This feature is particularly advantageous for training large models. In addition, we also demonstrate that \mavolion{} presents a more favorable performance-bandwidth balance compared to existing efficient distributed methods such as deep gradient compression and ternary gradients.

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ICLR Conference 2024 Conference Paper

Lion Secretly Solves a Constrained Optimization: As Lyapunov Predicts

Lizhang Chen
Bo Liu 0042
Kaizhao Liang
Qiang Liu 0001

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It achieves results comparable to AdamW but with greater memory efficiency. As what we can expect from the result of the random search, Lion blends a number of elements from existing algorithms, including signed momentum, decoupled weight decay, Polayk and Nesterov momentum, but doesn't fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This absence of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Using both continuous-time and discrete-time analysis, we demonstrate that Lion is a novel and theoretically grounded approach for minimizing a general loss function $f(x)$ while enforcing a bound constraint $||x||_\infty \leq 1/\lambda$. Lion achieves this through the incorporation of decoupled weight decay, where $\lambda$ represents the weight decay coefficient. Our analysis is facilitated by the development of a new Lyapunov function for the Lion updates. It applies to a wide range of Lion-$\phi$ algorithms, where the $sign(\cdot)$ operator in Lion is replaced by the subgradient of a convex function $\phi$, leading to the solution of the general composite optimization problem $\min_x f(x) + \phi^*(x)$. Our findings provide valuable insights into the dynamics of Lion and pave the way for further enhancements and extensions of Lion-related algorithms.

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NeurIPS Conference 2024 Conference Paper

Memory-Efficient LLM Training with Online Subspace Descent

Kaizhao Liang
Bo Liu
Lizhang Chen
Qiang Liu

Recently, a wide range of memory-efficient LLM training algorithms have gained substantial popularity. These methods leverage the low-rank structure of gradients to project optimizer states into a subspace using projection matrix found by singular value decomposition (SVD). However, convergence of these algorithms is highly dependent on the update rules of their projection matrix. In this work, we provide the \emph{first} convergence guarantee for arbitrary update rules of projection matrix. This guarantee is generally applicable to optimizers that can be analyzed with Hamiltonian Descent, including most common ones, such as LION, Adam. Inspired by our theoretical understanding, we propose Online Subspace Descent, a new family of subspace descent optimizer without SVD. Instead of updating projection matrix with eigenvectors, Online Subspace Descent updates projection matrix wtih online PCA. Online Subspace Descent is flexible and introduces only minimum overhead to training. We demonstrate that, for the task of pretraining LLaMA models ranging from 60M to 1B parameters on the C4 dataset, Online Subspace Descent achieves lower perplexity than state-of-the-art low-rank training methods across different settings and narrows the gap with full-rank baselines.

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