Lion-K Family of Optimizers

• Lion-K optimizers are a unified class of momentum-based methods built on convex analysis, norm constraints, and Lyapunov stability.
• They generalize the original Lion optimizer by varying the convex function or kinetic map to adapt to different geometric structures.
• Their design enables efficient applications in deep learning, image classification, language modeling, and robust distributed optimization.

The Lion-$\mathcal{K}$ family of optimizers encompasses a theoretically grounded and practically robust class of momentum-based optimization algorithms. This family generalizes the Lion optimizer (“Evolved Sign Momentum”) by subsuming it within a broader mathematical framework defined by convex analysis, norm constraints, and Lyapunov-based stability principles. Lion-$\mathcal{K}$ unifies a variety of well-known, new, and hybrid optimization techniques under a single algorithmic and theoretical perspective by systematically varying the choice of a convex function or kinetic map $\mathcal{K}$.

1. Theoretical Foundations and Update Structure

The Lion-$\mathcal{K}$ family is structured as a generalization of Lion, which itself was discovered through symbolic program search and is defined by momentum tracking and sign-based updates. The canonical discrete-time update of a Lion-$\mathcal{K}$ optimizer is given by: $$\begin{cases} m_{t+1} = \beta_2 m_t - (1-\beta_2)\nabla f(x_t) \ x_{t+1} = x_t + \epsilon \left( \partial\mathcal{K}\big( \beta_1 m_t - (1-\beta_1)\nabla f(x_t) \big) - \lambda x_t\right) \end{cases}$$ where $\partial\mathcal{K}$ denotes a (sub)gradient of the convex function $\mathcal{K}$, $\beta_1, \beta_2$ are momentum coefficients, $\lambda$ is the weight decay parameter, and $\epsilon$ is the learning rate (Chen et al., 2023).

When $\mathcal{K}(x) = \|x\|_1$, the standard Lion is recovered, with $\partial\mathcal{K}(x) = \mathrm{sign}(x)$. By choosing different $\mathcal{K}$, the update can project into, or regularize against, different geometric structures.

The continuous time analog is described by the ODE system: $$\begin{cases} \dot m_t = -\alpha \nabla f(x_t) - \gamma m_t \ \dot x_t = \partial\mathcal{K}(m_t - \varepsilon(\alpha \nabla f(x_t) + \gamma m_t)) - \lambda x_t \end{cases}$$ with tuning parameters $\alpha, \gamma, \lambda, \varepsilon > 0$ (Chen et al., 2023).

2. Composite Optimization and Norm-Constrained Perspectives

A central theoretical result is that Lion-$\mathcal{K}$ optimizers solve a composite optimization problem: $\min_{x}\ f(x) + \mathcal{K}^*(x)$ where $\mathcal{K}^*$ is the convex conjugate of $\mathcal{K}$, defined as $\mathcal{K}^*(x):=\sup_{z}(x^T z - \mathcal{K}(z))$ (Chen et al., 2023). For norm choices,

• $\mathcal{K}(x) = \|x\|_1$ leads to $\mathcal{K}^*(x)$ as the indicator function for $\|x\|_\infty \leq 1$, enforcing a box constraint on parameters.

The connection to constrained optimization is formalized through the Karush-Kuhn-Tucker (KKT) conditions, and the Lyapunov analysis demonstrates the optimizer’s two-phase convergence: rapid projection onto the feasible set and descent to composite minimizers (Chen et al., 2023). More generally, choosing $\mathcal{K}$ to be any norm or spectral function accommodates various constraints—$\ell_2$-norm, group norms, or spectral norms, as in the Muon variant (Chen et al., 18 Jun 2025).

3. Membership, Generalization, and Extensions

The Lion-$\mathcal{K}$ family includes and connects:

• Classical momenta (Polyak, Nesterov) with quadratic or mixed kinetic energy functions.
• Lion: using $\ell_1$ as above.
• Muon: using the nuclear norm, which, via its dual, enforces a spectral norm constraint (Chen et al., 18 Jun 2025).
• Frank-Wolfe/Mirror Descent: retrievable as specific $\mathcal{K}$ instances.

This framework provides a mechanism for incorporating new and hybrid forms of momentum or projection. For example, sorted or clipped norms induce hard and soft thresholding or sparsity-enforcing behaviors. The Lion-$\mathcal{K}$ class is further extensible to robust variants by integrating heavy-tailed gradient clipping, directly corresponding to robust Frank-Wolfe methods (Sfyraki et al., 4 Jun 2025).

A summary of possible choices and their corresponding behaviors:

$\mathcal{K}(x)$	$\partial \mathcal{K}(x)$	Constraint/Regularization
$\|x\|_1$	$\mathrm{sign}(x)$	$\|x\|_\infty \leq 1/\lambda$
$\|x\|_2$	$x/\|x\|_2$	$\|x\|_2 \leq 1/\lambda$
$\|X\|_*$ (nuclear norm)	matrix sign (SVD-based)	$\|X\| \leq 1/\lambda$ (spectral norm)

4. Convergence Properties and Robustness

Lion-$\mathcal{K}$ optimizers admit rigorous Lyapunov (energy) functions ensuring monotonic decrease and stability: $$H(x, m) = \alpha f(x) + \frac{\gamma}{\lambda} \mathcal{K}^*(\lambda x) + \frac{1-\varepsilon \gamma}{1+\varepsilon \lambda}\left[\mathcal{K}^*(\lambda x) + \mathcal{K}(m) - \lambda m^\top x\right]$$ (Chen et al., 2023).

Stopping criteria may be formulated in terms of generalized Frank-Wolfe gaps or $\ell_*$ norm gradients, coinciding with the KKT point of the underlying constrained problem (Sfyraki et al., 4 Jun 2025). For heavy-tailed noise regimes, robust Lion-$\mathcal{K}$ variants incorporating gradient clipping yield the best-known convergence rates for nonconvex stochastic optimization under minimal moment assumptions (Sfyraki et al., 4 Jun 2025).

5. Practical Instantiations and Applications

Lion-$\mathcal{K}$ family members are used in a range of modern machine learning tasks:

• Lion is applied to image classification with vision transformers, large-scale LLMing, vision-language contrastive learning, and diffusion models, showing state-of-the-art or highly competitive outcomes and superior memory/computation characteristics (Chen et al., 2023).
• Muon provides implicit spectral norm regularization for matrix weights, which has practical implications for stability and generalization, particularly in settings prone to spectral norm growth (e.g., deep networks, stabilizing GANs) (Chen et al., 18 Jun 2025).
• Robust versions (with clipping) are used to improve resilience to heavy-tailed gradient noise in deep LLMing and image classification, accelerating convergence and enhancing stability in high-dimensional or noisy regimes (Sfyraki et al., 4 Jun 2025).

Distributed and communication-efficient variants, such as Distributed Lion and Lion Cub, exploit the sign-based structure of updates (especially for $\mathcal{K}(x) = \|x\|_1$), leading to order-of-magnitude reductions in bandwidth while maintaining accuracy for vision and LLMs (Liu et al., 30 Mar 2024, Ishikawa et al., 25 Nov 2024).

6. Modern Extensions, Limitations, and Future Directions

Recent work proposes algorithmic extensions such as Cautious Lion (C-Lion), which masks updates misaligned with gradients, preserving monotonic loss descent and improving sample efficiency (Liang et al., 25 Nov 2024), as well as schedule-free and accelerated variants that dynamically interpolate between momentum behaviors (Morwani et al., 4 Feb 2025).

Information-theoretic analyses highlight the role of the entropy gap—a metric accounting for loss landscape conditioning and regularity—in understanding and improving generalization properties of optimizers in the Lion-$\mathcal{K}$ family (Tan et al., 28 Feb 2025).

Potential limitations, such as reduced gains over SGD/Adam in convolutional architectures or excessive regularization at small batch sizes, are documented (Chen et al., 2023). The design space and empirical behavior remain active research areas, particularly regarding:

• Selector/adaptive mechanisms for $\mathcal{K}$ targeting specific geometric or statistical priors
• Dynamic or learned mixture-of-$\mathcal{K}$ frameworks (multi-memory units, RLLC) for greater adaptability (Szegedy et al., 23 Feb 2024)
• Information-loss tradeoffs in the sign or quantization step, motivating soft-thresholded or non-binary update rules (Tan et al., 28 Feb 2025)

7. Summary Table: Representative Members and Key Properties

Optimizer	$\mathcal{K}$	Constraint/Reg.	Notable Application Contexts
Lion	$\|x\|_1$	$\|x\|_\infty \leq 1/\lambda$	Vision transformers, large-scale LLMs
Muon	$\|X\|_*$ (nuclear)	$\|X\| \leq 1/\lambda$	Spectral regularization, matrix weights
C-Lion	$\|x\|_1$, masked	Masked, sign aligned	LLMing, faster pretraining
Robust Lion	$\|x\|_1$, clipped	Clipped updates	Heavy-tailed noise, robust optimization
Distributed Lion, Lion Cub	$\|x\|_1$	Communication efficient	Large and distributed models, bandwidth-limited

Conclusion

The Lion-$\mathcal{K}$ family represents a theoretically principled and empirically validated approach to designing scalable, generalizable, and robust optimizers. By leveraging convex analysis, norm duality, Lyapunov stability, and geometric projections, it forms a foundational unification—encompassing Lion, Muon, and their modern variants—of optimization methods that actively shape parameter constraints, inductive bias, and information flow in state-of-the-art machine learning systems.