Lions and Muons: Optimization via Stochastic Frank-Wolfe under Heavy-Tailed Noise

Stochastic Frank-Wolfe is a classical optimization method for solving constrained optimization problems. On the other hand, recent optimizers such as Lion and Muon have gained quite significant popularity in deep learning. In this work, building on recent initiatives, we provide a unifying perspective by interpreting these seemingly disparate methods through the lens of Stochastic Frank-Wolfe. Specifically, we show that Lion and Muon with weight decay can be viewed as special instances of a Stochastic Frank-Wolfe, and we establish their convergence guarantees in terms of the Frank-Wolfe gap, a standard stationarity measure in non-convex optimization for Frank-Wolfe methods. We further find that convergence to this gap implies convergence to a KKT point of the original problem under a norm constraint for Lion and Muon. Moreover, motivated by recent empirical findings that stochastic gradients in modern machine learning tasks often exhibit heavy-tailed distributions, we extend Stochastic Frank-Wolfe to settings with heavy-tailed noise by developing two robust variants with strong theoretical guarantees that hold for general compact convex sets without the need for a large batch size, filling the gap in the literature on Stochastic Frank-Wolfe for non-convex optimization. Our contributions in the later part of this work, in turn, yield new variants of Lion and Muon, that better accommodate heavy-tailed gradient noise, thereby enhancing their practical scope.

Stochastic Frank-Wolfe is a classical optimization algorithm designed for problems with constraints on the solution. Lion and Muon, on the other hand, are two recently proposed optimizers that have shown strong empirical performance in training machine learning models. In this work, we show that Lion and Muon can be viewed as special cases of the Stochastic Frank-Wolfe algorithm. This connection offers a unified perspective on these methods, linking classical optimization theory to modern machine learning practice, and enables us to derive convergence guarantees for them. We also consider a practical challenge that arises in machine learning: gradient estimates can sometimes exhibit unusually large values, an effect known as heavy-tailed noise. To address this, we propose two variants of Stochastic Frank-Wolfe that can handle heavy-tailed noise more effectively. These algorithms, in turn, give rise to new variants of Lion and Muon that are more robust to heavy-tailed noise, making them better suited for modern machine learning setups.