The ever-growing scale of deep learning models and datasets underscores the critical importance of efficient optimization methods. While preconditioned gradient methods such as Adam and AdamW are the de facto optimizers for training neural networks and LLMs, structure-aware preconditioned optimizers like Shampoo and Muon, which utilize the matrix structure of gradients, have demonstrated promising evidence of faster convergence. In this talk, we introduce a unifying framework for analyzing “matrix-aware” preconditioned methods, which not only sheds light on the effectiveness of Muon and related optimizers but also leads to a class of new structure-aware preconditioned methods. A key contribution of this framework is its precise distinction between preconditioning strategies that treat neural network weights as vectors versus those that consider their matrix structure. This perspective provides new insights into several empirical phenomena in LLM pre-training, including Adam's training instabilities, Muon's accelerated convergence, and the necessity of learning rate warmup for Adam. Furthermore, we introduce PolarGrad, a new class of preconditioned optimization methods based on the polar decomposition of matrix-valued gradients. PolarGrad includes Muon as a special instance. Our extensive evaluations across diverse matrix optimization problems and language model pre-training tasks demonstrate that PolarGrad outperforms both Adam and Muon.

Associate Professor of Department of Mathematics and Wharton Statistics and Data Science Department of University of Pennsylvania, Co-Director Penn Research in Machine Learning. His research interests: mathematical theory of  deep learning and AI, statistical foundations of large language models, privacy-preserving machine learning, high-dimensional statistics, mathematical optimization.

He is fellow of IMS, and received Sloan Research Fellowship, SIAM Early Career Prize in Data Science, and NSF CAREER Award. He is on editorial board of Journal of Machine Learning Research, Journal of the American Statistical Association, and Operations Research et al.