Papers › Adam Exploits ℓ_∞-geometry of Loss Landscape via Coordinate-wise Adaptivity

Adam Exploits ℓ_∞-geometry of Loss Landscape via Coordinate-wise Adaptivity

10 Oct 2024arXiv:2410.08198archive 2025-07-28

Shuo Xie, Mohamad Amin Mohamadi, Zhiyuan Li

Adam outperforms SGD when training language models. Yet this advantage is not well-understood theoretically -- previous convergence analysis for Adam and SGD mainly focuses on the number of steps T and is already minimax-optimal in non-convex cases, which are both O(T^(-1/4)). In this work, we argue that the exploitation of nice ℓ_∞-geometry is the key advantage of Adam over SGD. More specifically, we give a new convergence analysis for Adam under novel assumptions that loss is smooth under ℓ_∞-geometry rather than the more common ℓ₂-geometry, which yields a much better empirical smoothness constant for GPT-2 and ResNet models. Our experiments confirm that Adam performs much worse when the favorable ℓ_∞-geometry is changed while SGD provably remains unaffected. We also extend the convergence analysis to blockwise Adam under novel blockwise smoothness assumptions.

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