Convergence Rate Analysis of the AdamW-Style Shampoo: Unifying One-Sided and Two-Sided Preconditioning

This paper studies the AdamW-style Shampoo optimizer, an effective implementation of the 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 $\frac{1}{K}\sum_{k=1}^KE[|||\nabla f(X_k)|||]\leq O(\frac{\sqrt{m+n}C}{K^{1/4}})$ measured by nuclear norm (denoted as $|||\cdot|||$ to display correctly in OpenReview), 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 $||\nabla f(X)||\leq|||\nabla f(X)|||\leq\sqrt{\min(m,n)}||\nabla f(X)||$ (denote $||\cdot||$ as the Frobenius norm to display correctly in OpenReview), supporting that our convergence rate can be considered to be analogous to the optimal $\frac{1}{K}\sum_{k=1}^K E[||\nabla f(X_k)||]\leq O(\frac{C}{K^{1/4}})$ convergence rate of SGD in the ideal case of $|||\nabla f(X)|||= \Theta(\sqrt{\min(m,n)})||\nabla f(X)||$ and balanced $m$ and $n$.

The AdamW-style Shampoo optimizer is an effective implementation of the classical Shampoo that notably won the external tuning track of the AlgoPerf neural network training algorithm competition. However, its convergence behavior is not well-understood. We establish the convergence rate that can be considered to be analogous to the optimal convergence rate of SGD in the ideal case. Our analysis unifies one-sided and two-sided preconditioning. Our work provides theoretical support for the convergence of the AdamW-style Shampoo optimizer when training deep neural networks.