New Study Provides Convergence Guarantees for AdamW-Style Shampoo Optimizer

Global: New Study Provides Convergence Guarantees for AdamW-Style Shampoo Optimizer

Researchers Huan Li, Yiming Dong, and Zhouchen Lin submitted a paper on January 12, 2026, that delivers a formal convergence‑rate analysis for the AdamW‑style Shampoo optimizer, an algorithm that previously earned top honors in the external tuning track of the AlgoPerf neural network training competition.

Background

The Shampoo optimizer, originally introduced as a second‑order method for deep learning, has been adapted to an AdamW‑style variant that incorporates weight‑decay regularization. Its practical success in benchmark competitions has spurred interest in understanding its theoretical properties, a gap the new work seeks to fill.

Unified Preconditioning Framework

The authors propose a unified analytical framework that bridges one‑sided and two‑sided preconditioning techniques. By treating both forms within a single mathematical construct, the paper streamlines prior disparate analyses and establishes a common ground for evaluating optimizer behavior.

Convergence Rate Findings

Within this framework, the study proves that the expected nuclear‑norm gradient magnitude satisfies (frac{1}{K}sum_{k=1}^{K}mathbb{E}[|nabla f(X_k)|_*] leq Oleft(frac{sqrt{m+n},C}{K^{1/4}}right)), where (K) denotes the iteration count, ((m,n)) the dimensions of matrix‑valued parameters, and (C) a constant matching that of the optimal stochastic gradient descent (SGD) rate. The analysis also confirms the inequalities (|nabla f(X)|_F leq |nabla f(X)|_* leq sqrt{m+n},|nabla f(X)|_F), linking the nuclear norm to the more familiar Frobenius norm.

Comparison with Stochastic Gradient Descent

Because the derived bound mirrors the optimal SGD rate of (O(C/K^{1/4})) when (|nabla f(X)|_* = Theta(sqrt{m+n}),|nabla f(X)|_F), the authors argue that the AdamW‑style Shampoo optimizer attains comparable theoretical efficiency while offering the practical benefits of adaptive preconditioning.

Implications for Large‑Scale Training

The results suggest that practitioners training large neural networks may employ the AdamW‑style Shampoo optimizer without sacrificing asymptotic convergence speed. The nuclear‑norm perspective also highlights potential advantages in handling high‑dimensional parameter matrices common in modern architectures.

Future Directions

The paper acknowledges that the analysis assumes certain smoothness conditions and that empirical validation on diverse workloads remains an open avenue. Further research may explore extensions to non‑convex settings and integration with distributed training pipelines.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.