Adaptive gradient algorithms have been widely adopted in training large-scale deep neural networks, especially large foundation models. Despite their huge success in practice, their theoretical advantages over stochastic gradient descent (SGD) have not been fully understood, especially in the large batch-size setting commonly used in practice. This is because the only theoretical result that can demonstrate the benefit of Adagrad over SGD was obtained in the original paper of Adagrad for nonsmooth objective functions. However, for nonsmooth objective functions, there can be a linear slowdown of convergence when batch size increases, and thus a convergence analysis based on nonsmooth assumption cannot be used for large batch algorithms. In this work, we resolve this gap between theory and practice by providing a new analysis of Adagrad on both convex and nonconvex smooth objectives suitable for the large batch setting. It is shown that under the anisotropic smoothness and noise conditions, increased batch size does not slow down convergence for Adagrad, and thus it can still achieve a faster convergence guarantee over SGD even in the large batch setting. We present detailed comparisons between SGD and Adagrad to provide a better understanding of the benefits of adaptive gradient methods. Experiments in logistic regression and instruction following fine-tuning tasks provide strong evidence to support our theoretical analysis.

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