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arXiv · 2310.17998

Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity

Bohan Wang·Jingwen Fu·Huishuai Zhang·Nanning Zheng·Wei Chen
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Abstract

Recently, Arjevani et al. [1] established a lower bound of iteration complexity for the first-order optimization under an $L$-smooth condition and a bounded noise variance assumption. However, a thorough review of existing literature on Adam's convergence reveals a noticeable gap: none of them meet the above lower bound. In this paper, we close the gap by deriving a new convergence guarantee of Adam, with only an $L$-smooth condition and a bounded noise variance assumption. Our results remain valid across a broad spectrum of hyperparameters. Especially with properly chosen hyperparameters, we derive an upper bound of the iteration complexity of Adam and show that it meets the lower bound for first-order optimizers. To the best of our knowledge, this is the first to establish such a tight upper bound for Adam's convergence. Our proof utilizes novel techniques to handle the entanglement between momentum and adaptive learning rate and to convert the first-order term in the Descent Lemma to the gradient norm, which may be of independent interest.

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Bohan Wang, Jingwen Fu, Huishuai Zhang, Nanning Zheng, Wei Chen. 2023-10-27. Closing the Gap Between the Upper Bound and the Lower Bound of Adam's Iteration Complexity. https://arxiv.org/abs/2310.17998

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