Adam Exploits $\ell_\infty$-geometry of Loss Landscape via Coordinate-wise Adaptivity

• URL: http://arxiv.org/abs/2410.08198v1
• Date: Thu, 10 Oct 2024 17:58:53 GMT
• Title: Adam Exploits $\ell_\infty$-geometry of Loss Landscape via Coordinate-wise Adaptivity
• Authors: Shuo Xie, Mohamad Amin Mohamadi, Zhiyuan Li,
• Abstract summary: We find that Adam performs worse when the favorable $ell_infty$-geometry is SGD while provably remains unaffected. Our experiments confirm that Adam performs much worse when the favorable $ell_infty$-geometry is SGD while provably remains unaffected.
• Score: 6.270305440413688
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\widetilde{O}(T^{-1/4})$. In this work, we argue that the exploitation of nice $\ell_\infty$-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 $\ell_\infty$-geometry rather than the more common $\ell_2$-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 $\ell_\infty$-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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