Optimal Growth Schedules for Batch Size and Learning Rate in SGD that Reduce SFO Complexity

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.