Accelerating LMO-Based Optimization via Implicit Gradient Transport

• URL: http://arxiv.org/abs/2605.05577v1
• Date: Thu, 07 May 2026 01:44:57 GMT
• Title: Accelerating LMO-Based Optimization via Implicit Gradient Transport
• Abstract summary: We propose emphLMO-IGT, a new class of normalizing LMO-based methods leveraging implicit gradient transport (IGT)<n>By evaluating gradients at transported points, LMO-IGT accelerates convergence while retaining the single-gradient-per-iteration structure of standard LMO.<n>Our analysis establishes that LMO achieves an complexity of $mathcalO(varepsilon-4)$, variance-reduced LMO achieves $mathcalO(varepsilon-3)$ at the cost additional gradient evaluations, and
• Score: 10.973489259319841
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent optimizers such as Lion and Muon have demonstrated strong empirical performance by normalizing gradient momentum via linear minimization oracles (LMOs). While variance reduction has been explored to accelerate LMO-based methods, it typically incurs substantial computational overhead due to additional gradient evaluations. At the same time, the theoretical understanding of LMO-based methods remains fragmented across unconstrained and constrained formulations. Motivated by these limitations, we propose \emph{LMO-IGT}, a new class of stochastic LMO-based methods leveraging implicit gradient transport (IGT). We further introduce a unified framework for stochastic LMO-based optimization together with a new stationarity measure, the \emph{regularized support function} (RSF), which bridges gradient-norm and Frank--Wolfe-gap notions within a common framework. By evaluating stochastic gradients at transported points, LMO-IGT accelerates convergence while retaining the single-gradient-per-iteration structure of standard stochastic LMO. Our analysis establishes that stochastic LMO achieves an iteration complexity of $\mathcal{O}(\varepsilon^{-4})$, variance-reduced LMO achieves $\mathcal{O}(\varepsilon^{-3})$ at the cost of additional gradient evaluations, and LMO-IGT achieves $\mathcal{O}(\varepsilon^{-3.5})$ using only a single stochastic gradient per iteration. Empirically, LMO-IGT consistently improves over stochastic LMO counterparts with negligible overhead. Among its instantiations, Muon-IGT achieves the strongest overall performance across evaluated settings, demonstrating that IGT provides an effective and practical acceleration mechanism for modern LMO-based optimization.

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