Accelerating LMO-Based Optimization via Implicit Gradient Transport

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.

Provably Near-Optimal Federated Ensemble Distillation with Negligible Overhead

Federated ensemble distillation addresses client heterogeneity by generating pseudo-labels for an unlabeled server dataset based on client predictions and training the server model using the pseudo-labeled dataset. The unlabeled server dataset can either be pre-existing or generated through a data-free approach. The effectiveness of this approach critically depends on the method of assigning weights to client predictions when creating pseudo-labels, especially in highly heterogeneous settings. Inspired by theoretical results from GANs, we propose a provably near-optimal weighting method that leverages client discriminators trained with a server-distributed generator and local datasets. Our experiments on various image classification tasks demonstrate that the proposed method significantly outperforms baselines. Furthermore, we show that the additional communication cost, client-side privacy leakage, and client-side computational overhead introduced by our method are negligible, both in scenarios with and without a pre-existing server dataset.