Lizhang Chen

Lizhang Chen, Bo Liu, Kaizhao Liang et al.

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It performs comparably or favorably to AdamW but with greater memory efficiency. As we can expect from the results of a random search program, Lion incorporates elements from several existing algorithms, including signed momentum, decoupled weight decay, Polak, and Nesterov momentum, but does not 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 lack of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Based on both continuous-time and discrete-time analysis, we demonstrate that Lion is a theoretically novel and principled approach for minimizing a general loss function $f(x)$ while enforcing a bound constraint $\|x\|_\infty \leq 1/λ$. Lion achieves this through the incorporation of decoupled weight decay, where $λ$ represents the weight decay coefficient. Our analysis is made possible by the development of a new Lyapunov function for the Lion updates. It applies to a broader family of Lion-$κ$ algorithms, where the $\text{sign}(\cdot)$ operator in Lion is replaced by the subgradient of a convex function $κ$, leading to the solution of a general composite optimization problem of $\min_x f(x) + κ^*(x)$. Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms.

Kaizhao Liang, Bo Liu, Lizhang Chen et al.

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 the projection matrix with eigenvectors, Online Subspace Descent updates the projection matrix with online PCA. Online Subspace Descent is flexible and introduces only minimum overhead to training. We show that for the task of pretraining LLaMA models ranging from 60M to 7B parameters on the C4 dataset, Online Subspace Descent achieves lower perplexity and better downstream tasks performance than state-of-the-art low-rank training methods across different settings and narrows the gap with full-rank baselines.

Kaizhao Liang, Lizhang Chen, Bo Liu et al.

AdamW has been the default optimizer for transformer pretraining. For many years, our community searched for faster and more stable optimizers with only constrained positive outcomes. In this work, we propose a single-line modification in Pytorch to any momentum-based optimizer, which we rename cautious optimizer, e.g. C-AdamW and C-Lion. Our theoretical result shows that this modification preserves Adam's Hamiltonian function and it does not break the convergence guarantee under the Lyapunov analysis. In addition, a whole new family of optimizers is revealed by our theoretical insight. Among them, we pick the simplest one for empirical experiments, showing not only speed-up on Llama and MAE pretraining up to $1.47$ times, but also better results in LLM post-training tasks. Code is available at https://github.com/kyleliang919/C-Optim.

Bo Liu, Lemeng Wu, Lizhang Chen et al.

The Lion optimizer has been a promising competitor with the AdamW for training large AI models, with advantages on 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 communicating binary or lower-precision vectors between 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 Distributed Lion presents a more favorable performance-bandwidth balance compared to existing efficient distributed methods such as deep gradient compression and ternary gradients.

Lizhang Chen, Jonathan Li, Kaizhao Liang et al.

We introduce Cautious Weight Decay (CWD), a one-line, optimizer-agnostic modification that applies weight decay only to parameter coordinates whose signs align with the optimizer update. Unlike standard decoupled decay, which implicitly optimizes a regularized or constrained objective, CWD preserves the original loss and admits a bilevel interpretation: it induces sliding-mode behavior upon reaching the stationary manifold, allowing it to search for locally Pareto-optimal stationary points of the unmodified objective. In practice, CWD is a drop-in change for optimizers such as AdamW, Lion, and Muon, requiring no new hyperparameters or additional tuning. For language model pre-training and ImageNet classification, CWD consistently improves final loss and accuracy at million- to billion-parameter scales.