As training data become larger, distributed training has been broadly adopted in machine learning tasks.

Muon, a recently proposed optimizer that leverages the inherent matrix structure of neural network parameters, has demonstrated strong empirical performance, indicating its potential as a successor to standard optimizers such as AdamW. This paper presents theoretical analysis to support its practical success. We provide convergence proofs for Muon across four practical settings, systematically examining its behavior with and without the inclusion of Nesterov momentum and weight decay. Our analysis covers the standard configuration using both, thereby elucidating its real-world performance. We then demonstrate that the addition of weight decay yields strictly tighter theoretical bounds and clarify the interplay between the weight decay coefficient and the learning rate. Finally, we derive the critical batch size for Muon that minimizes the computational cost of training. Our analysis identifies the hyperparameters governing this value, and our experiments validate the corresponding theoretical findings across workloads including image classification and language modeling task.

Equal Contribution 32nd Conference on Neural Information Processing Systems (NeurIPS 2018), Montréal, Canada. is often attributed to the rapid decay in the learning rate when gradients are dense, which is often the case in many machine learning applications.

As training data become larger, distributed training has been broadly adopted in machine learning tasks.

This paper considers stochastic first-order algorithms for minimax optimization under Polyak-Łojasiewicz (PL) conditions.

The convergence behavior of mini-batch stochastic gradient descent (SGD) is highly sensitive to the batch size and learning rate settings. Recent theoretical studies have identified the existence of a critical batch size that minimizes stochastic first-order oracle (SFO) complexity, defined as the expected number of gradient evaluations required to reach a stationary point of the empirical loss function in a deep neural network. An adaptive scheduling strategy is introduced to accelerate SGD that leverages theoretical findings on the critical batch size. The batch size and learning rate are adjusted on the basis of the observed decay in the full gradient norm during training. Experiments using an adaptive joint scheduler based on this strategy demonstrated improved convergence speed compared with that of existing schedulers.

The unprecedented growth of deep learning models has enabled remarkable advances but introduced substantial computational bottlenecks. A key factor contributing to training efficiency is batch-size and learning-rate scheduling in stochastic gradient methods. However, naive scheduling of these hyperparameters can degrade optimization efficiency and compromise generalization. Motivated by recent theoretical insights, we investigated how the batch size and learning rate should be increased during training to balance efficiency and convergence. We analyzed this problem on the basis of stochastic first-order oracle (SFO) complexity, defined as the expected number of gradient evaluations needed to reach an $ε$-approximate stationary point of the empirical loss. We theoretically derived optimal growth schedules for the batch size and learning rate that reduce SFO complexity and validated them through extensive experiments. Our results offer both theoretical insights and practical guidelines for scalable and efficient large-batch training in deep learning.