🤖 AI Summary
The Lion optimizer (Evolved Sign Momentum) lacks a rigorous theoretical foundation. Method: This work establishes that Lion implicitly solves the constrained optimization problem $min_x f(x)$ subject to $|x|_infty leq 1/lambda$, achieving implicit $ell_infty$-norm regularization via decoupled weight decay. We construct the first Lyapunov function tailored to Lion’s update rule and rigorously prove its convergence and equivalence to constrained optimization. Furthermore, we generalize Lion to the Lion-$kappa$ family, interpreting the sign operator as a subgradient of a convex function $kappa$, thereby unifying diverse variants. Our analysis integrates continuous and discrete dynamical systems modeling, Lyapunov stability theory, and subgradient optimization. Contribution/Results: This work provides the first complete theoretical framework for Lion—elevating it from a heuristic method to a principled algorithm for bounded constraint optimization—and lays a solid mathematical foundation for performance enhancement and generalization.
📝 Abstract
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 $|x|_infty leq 1/lambda$. Lion achieves this through the incorporation of decoupled weight decay, where $lambda$ 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-$kappa$ algorithms, where the $ ext{sign}(cdot)$ operator in Lion is replaced by the subgradient of a convex function $kappa$, leading to the solution of a general composite optimization problem of $min_x f(x) + kappa^*(x)$. Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms.
