# DASH: Faster Shampoo via Batched Block Preconditioning and Efficient Inverse-Root Solvers

arXiv: https://arxiv.org/abs/2602.02016 · HTML: https://mlsanity.com/paper/2602.02016/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 32 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Builds on (most cited first, 16 of 16)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [ZeRO: Memory Optimizations Toward Training Trillion Parameter Models](https://mlsanity.com/paper/1910.02054/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [A Distributed Data-Parallel PyTorch Implementation of the Distributed Shampoo Optimizer for Training Neural Networks At-Scale](https://mlsanity.com/paper/2309.06497/) (2023)
- [Benchmarking Neural Network Training Algorithms](https://mlsanity.com/paper/2306.07179/) (2023)
- [Disentangling Adaptive Gradient Methods from Learning Rates](https://mlsanity.com/paper/2002.11803/) (2020)
- [Structured Preconditioners in Adaptive Optimization: A Unified Analysis](https://mlsanity.com/paper/2503.10537/) (2025)
- [Purifying Shampoo: Investigating Shampoo's Heuristics by Decomposing its Preconditioner](https://mlsanity.com/paper/2506.03595/) (2025)
- [4-bit Shampoo for Memory-Efficient Network Training](https://mlsanity.com/paper/2405.18144/) (2024)
- [Optimizers Qualitatively Alter Solutions And We Should Leverage This](https://mlsanity.com/paper/2507.12224/) (2025)
- [Understanding and Improving Shampoo and SOAP via Kullback-Leibler Minimization](https://mlsanity.com/paper/2509.03378/) (2025)
- [Error Feedback Can Accurately Compress Preconditioners](https://mlsanity.com/paper/2306.06098/) (2023)
- [Beyond Outliers: A Study of Optimizers Under Quantization](https://mlsanity.com/paper/2509.23500/) (2025)
- [Memory-Efficient 4-bit Preconditioned Stochastic Optimization](https://mlsanity.com/paper/2412.10663/) (2024)

## Built on by (most cited first, 2 of 2)

- [SOAP, Muon, and Beyond: Pushing LLM Pretraining Scales](https://mlsanity.com/paper/2607.20548/) (2026)
- [Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization](https://mlsanity.com/paper/2610.16923/) (2026)

## Abstract (arXiv)

Shampoo is one of the leading approximate second-order optimizers: a variant of it has won the MLCommons AlgoPerf competition, and it has been shown to produce models with lower activation outliers that are easier to compress. Yet, applying Shampoo currently comes at the cost of significant computational slowdown, due to its expensive internal operations. In this paper, we take a significant step to address this shortcoming by proposing method (for Distributed Accelerated SHampoo), a faster implementation of Distributed Shampoo based on two main new techniques: First, we show that preconditioner blocks can be stacked into 3D tensors to significantly improve GPU utilization; second, we introduce the Newton-DB iteration and the Chebyshev polynomial approximations as novel and faster approaches for computing the inverse matrix roots required by Shampoo. Along with these algorithmic contributions, we provide a first in-depth analysis of how matrix scaling critically affects Shampoo convergence. On the practical side, our GPU-aware implementation achieves up to 5.6× faster optimizer steps compared to the well-optimized Distributed Shampoo, while Newton-DB attains the lowest validation perplexity per iteration among all tested methods. Our code is available at https://github.com/IST-DASLab/DASH.