Stochastic Gradient Methods with Online Scaling

Abstract: This paper introduces Stochastic Online Scaled Gradient Methods (SOSGM), a generalization of the recently developed adaptive preconditioning framework in arXiv:2505.23081 and arXiv:2509.11007 to stochastic optimization. Under standard assumptions, we establish convergence guarantees for SOSGM using large batchsize or variance reduction. SOSGM is compatible with popular diagonal and/or low-rank preconditioners as well as heavy-ball momentum, while maintaining memory and computation cost comparable to Adam. Extensive numerical experiments demonstrate the strong empirical performance of SOSGM. Using a diagonal preconditioner, SOSGM and its variants substantially outperform existing adaptive first-order methods across a range of statistical learning tasks.