Lion-𝒦 Framework: Geometry & Optimization

• Lion-𝒦 framework is a generalization that integrates pursuit-evasion dynamics with constrained optimization through structure-inducing convex sets, enabling robust multiagent coordination.
• The framework employs advance and cone moves to maintain the k-hull condition, ensuring finite-step capture with computable performance bounds.
• It extends to composite optimization, leveraging decoupled weight decay and Lyapunov functions to guarantee convergence in both centralized and communication-efficient distributed settings.

The Lion-$\mathcal{K}$ framework denotes a generalization of classical pursuit-evasion dynamics and constrained optimization, where coordination, geometric constraints, and composite objective minimization are governed by structure-inducing convex sets or functions $\mathcal{K}$. This framework originated from discrete multiagent pursuit-evasion (as formalized in the "k-capture" problem (Bopardikar et al., 2011)) and has been modernized for optimization through analysis of algorithms such as Lion (Evolved Sign Momentum) and Muon, both of which have been shown to solve constrained problems corresponding to specific choices of the $\mathcal{K}$-operator. The Lion-$\mathcal{K}$ paradigm can encode geometric, algebraic, or communication constraints and leads to provable guarantees in both game-theoretic pursuit and optimization settings.

1. Geometric Roots: k-Capture and the k-Hull Condition

The classical Lion-$\mathcal{K}$ framework arises from pursuit-evasion games in continuous Euclidean space, specifically the generalized $k$-capture problem (Bopardikar et al., 2011). In this setting, $n$ pursuers and one evader move in discrete time with equal maximum speed. The evader is captured only if at least $k$ pursuers simultaneously reach its exact position. The central geometric concept is the $k$-hull:

• $k$-Hull Definition: For a set $S$ of positions, Hull$_k(S)$ is the set of points $p$ such that, for any hyperplane $\ell$ through $p$, each closed half-space determined by $\ell$ contains at least $k$ members of $S$. Formally,

$$p \in \mathrm{Hull}_k(S) \iff \forall \ell \ni p,\, |S \cap (\text{half-space}_1)| \geq k \text{ and } |S \cap (\text{half-space}_2)| \geq k$$

Capture is possible if and only if the evader is in the strict interior of the pursuers’ $k$-hull at some time. This invariant enables multiagent coordination by geometric constraints and can be generalized through convex mappings $\mathcal{K}$ in higher-level frameworks.

2. Strategy Design: Advance and Cone Moves

Algorithmically, the Lion-$\mathcal{K}$ framework employs constructive strategies:

• Advance Move: Applied by pursuers not among the $k$ closest to the evader; they move along the ray parallel to their previous displacement relative to the evader, choosing the location minimizing their post-move distance.
• Cone Move: When $k$ pursuers are closest and the angle between their difference vectors and the evader’s movement is suitably bounded, these pursuers execute a cone move. The cone is anchored at the evader’s location, oriented along its velocity, and has half-angle $\beta_{\max}$, determined by

$$\beta_{\max} = \arccos \left( \min_{u_e \in S} g(u_e) \right)$$

where $g(u_e)$ is the $k$th maximum of $\{\cos\theta_j\}$ over pursuers $j$, with $\theta_j$ the angle between $p_j-e$ and the evader’s velocity $u_e$.

These strategies guarantee preservation of the $k$-hull condition, ensure safe progress for pursuers, and terminate in finite steps—specifically, in $n\left(1 + d_{\max}/\cos\beta_{\max}\right)^2$ moves, where $d_{\max}$ is the largest initial distance (see (Bopardikar et al., 2011)).

3. Composite Optimization Interpretation

The Lion-$\mathcal{K}$ framework has received new significance as an abstraction for constrained and composite optimization algorithms. The Lion optimizer (sign momentum, decoupled weight decay) is shown to minimize objectives of the form

$\min_{x} f(x) + \mathcal{K}^*(x)$

where $\mathcal{K}^*$ is the convex conjugate of the chosen function $\mathcal{K}$ (Chen et al., 2023).

Key cases:

• With $\mathcal{K}(x) = \|x\|_1$, the conjugate $\mathcal{K}^*(x)$ gives an indicator for the $\ell_\infty$ ball; Lion solves $f(x)$ subject to $\|x\|_\infty \leq 1/\lambda$.
• For matrix variables and $\mathcal{K}(X) = \|X\|_*$ (nuclear norm), Muon enforces a constraint on spectral norm: $f(X)$ subject to $\|X\|_\infty \leq 1/\lambda$ (Chen et al., 18 Jun 2025).

Decoupled weight decay ensures that the regularization or constraint is imposed independently from the momentum, which is crucial both for theory (Lyapunov function analysis) and for practical performance.

4. Theoretical Analysis and Lyapunov Functions

Continuous and discrete-time analyses rely on Lyapunov functions adapted to the Lion-$\mathcal{K}$ framework:

$$H(x, m) = \alpha f(x) + (\gamma/\lambda) \mathcal{K}^*(\lambda x) + \phi(m) - \lambda m^T x$$

with $H(x, m)$ non-increasing along trajectories, guaranteeing convergence to stationary points of the constrained or regularized problem (Chen et al., 2023). If the iterates leave the feasible set defined by $\mathcal{K}^*$, the penalty term decays the infeasibility exponentially fast,

$$\mathrm{dist}(\lambda x_t, \text{dom}~\mathcal{K}^*) \le \exp(-\lambda (t-s))\, \mathrm{dist}(\lambda x_s, \text{dom}~\mathcal{K}^*)$$

Under standard smoothness and bounded variance assumptions, convergence to a Karush-Kuhn-Tucker (KKT) point is established, e.g., $S(X_t) = \nabla F(X_t) + X\nabla F(X_t) \to 0$ at rate $O(1/\sqrt{T})$ for Muon (Chen et al., 18 Jun 2025). For scalar Lion, analogous results apply to coordinatewise sign and momentum.

5. Distributed and Communication-Efficient Extensions

The Lion-$\mathcal{K}$ framework admits scalable, communication-efficient adaptations:

• In federated learning, FedLion applies sign-compressed momenta and local model quantization, reducing uplink bandwidth to $O(\log(2E+1))$ bits per element (Tang et al., 15 Feb 2024). Global updates aggregate signed differences, and theoretical analysis confirms accelerated convergence, especially for dense gradients compared to FedAvg.
• Distributed Lion variants with unbiased sign compression maintain convergence guarantees even with 1-bit bidirectional communication, achieving rates such as $O(d^{1/4}T^{-1/4})$ for the gradient norm (Jiang et al., 17 Aug 2025).

These developments are consistent with $\mathcal{K}$ as an "editor's term" for communication/constraint encoding in distributed systems.

6. Connections to Metric Geometry and Quantitative Pursuit

Theoretical underpinnings expand beyond Euclidean spaces to general metric and geodesic settings. The uniform betweenness property (UBW) and related convexity concepts serve as moduli to extract explicit rates of capture and convergence in pursuit-evasion games (Kohlenbach et al., 2018). In bounded domains satisfying UBW, the Lion always wins, and proof mining enables uniform rates of convergence. This abstraction extends the Lion-$\mathcal{K}$ paradigm to domains with more general convex or curvature properties, yielding universal recipes for algorithmic design and performance prediction.

7. Implications and Generalizations

The Lion-$\mathcal{K}$ framework synthesizes strategies for k-agent pursuit, geometric constraint enforcement, composite optimization by regularization and decoupled penalty, and communication-efficient distributed coordination. Selection of $\mathcal{K}$ (norm, group norm, spectral function, quantization operator) tailors the optimization trajectory, constraint set, and communication properties, with rigorous guarantees rooted in Lyapunov function analysis, KKT theory, and metric geometry. The framework generalizes many classical methods (polyak, SGD, AdamW) and unifies them under a geometric and functional analytic lens, revealing systematic mechanisms for regularization and efficiency in modern deep learning and multiagent control.