Convergence Rate Analysis of the AdamW-Style Shampoo: Unifying One-Sided and Two-Sided Preconditioning
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| Format: | Preprint |
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2026
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| _version_ | 1866914598573047808 |
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| author | Li, Huan Dong, Yiming Lin, Zhouchen |
| author_facet | Li, Huan Dong, Yiming Lin, Zhouchen |
| contents | This paper studies the AdamW-style Shampoo optimizer, an effective implementation of classical Shampoo that notably won the external tuning track of the AlgoPerf neural network training algorithm competition. Our analysis unifies one-sided and two-sided preconditioning and establishes the convergence rate $\frac{1}{K}\sum_{k=1}^K E\left[\|\nabla f(X_k)\|_*\right]\leq O(\frac{\sqrt{m+n}C}{K^{1/4}})$ measured by nuclear norm, where $K$ represents the iteration number, $(m,n)$ denotes the size of matrix parameters, and $C$ matches the constant in the optimal convergence rate of SGD. Theoretically, we have $\|\nabla f(X)\|_F\leq \|\nabla f(X)\|_*\leq \sqrt{m+n}\|\nabla f(X)\|_F$, supporting that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[\|\nabla f(X_k)\|_F\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case of $\|\nabla f(X)\|_*= Θ(\sqrt{m+n})\|\nabla f(X)\|_F$ and balanced $m$ and $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_07326 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence Rate Analysis of the AdamW-Style Shampoo: Unifying One-Sided and Two-Sided Preconditioning Li, Huan Dong, Yiming Lin, Zhouchen Optimization and Control Machine Learning This paper studies the AdamW-style Shampoo optimizer, an effective implementation of classical Shampoo that notably won the external tuning track of the AlgoPerf neural network training algorithm competition. Our analysis unifies one-sided and two-sided preconditioning and establishes the convergence rate $\frac{1}{K}\sum_{k=1}^K E\left[\|\nabla f(X_k)\|_*\right]\leq O(\frac{\sqrt{m+n}C}{K^{1/4}})$ measured by nuclear norm, where $K$ represents the iteration number, $(m,n)$ denotes the size of matrix parameters, and $C$ matches the constant in the optimal convergence rate of SGD. Theoretically, we have $\|\nabla f(X)\|_F\leq \|\nabla f(X)\|_*\leq \sqrt{m+n}\|\nabla f(X)\|_F$, supporting that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^KE\left[\|\nabla f(X_k)\|_F\right]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case of $\|\nabla f(X)\|_*= Θ(\sqrt{m+n})\|\nabla f(X)\|_F$ and balanced $m$ and $n$. |
| title | Convergence Rate Analysis of the AdamW-Style Shampoo: Unifying One-Sided and Two-Sided Preconditioning |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2601.07326 |