Gram Newton-Schulz: A Fast, Hardware-Aware Newton-Schulz Algorithm for Muon

Authors: Jack Zhang, Noah Amsel, Berlin Chen, Tri Dao
Blogpost: https://dao-ailab.github.io/blog/2026/gram-newton-schulz/

Achieve up to 2x faster Newton-Schulz with Gram Newton-Schulz and symmetric CuTeDSL GEMM kernels!

What you're probably here for:

1. Gram Newton-Schulz: https://github.com/Dao-AILab/gram-newton-schulz/blob/main/gram_newton_schulz/gram_newton_schulz.py
2. Gram Newton-Schulz Restart Autotune: https://github.com/Dao-AILab/gram-newton-schulz/blob/main/gram_newton_schulz/restart_autotune.py
3. Symmetric GEMMs for Hopper and Blackwell in CuTeDSL: https://github.com/Dao-AILab/quack/blob/main/quack/gemm_symmetric.py

About

Gram Newton-Schulz is a hardware-aware algorithm for polar decomposition that is mathematically equivalent to and faster than Newton-Schulz. Polar decomposition is most commonly used in Muon, and Gram Newton-Schulz serves as a direct drop-in for standard Newton-Schulz with no training accuracy tradeoff.

Instead of iterating on the expensive $X \in \mathbb{R}^{n \times m}$ matrix, Gram Newton-Schulz iterates on the small, square, symmetric Gram matrix $XX^\top \in \mathbb{R}^{n \times n}$, lowering FLOPs and enabling more symmetric GEMM kernels.

Gram Newton-Schulz

Input: $X \in \mathbb{R}^{n \times m}$ with $n \leq m$, coefficients ${(a_t, b_t, c_t)}_{t=1}^5$

1. $X \gets X / (\|X\|_{F} + \epsilon)$   // Normalize sing vals to $[0, 1]$.   $\epsilon = 10^{-7}$
2. $X \gets \texttt{float16}(X)$   // Cast to half precision for speed
3. If $m < n$:   $X \gets X^\top$   // Trick to make $XX^\top$ cheaper
4. $R_0 \gets XX^\top$
5. $Q_0 \gets I$
6. For $t = 1, \ldots, 5$:
   • If $t = 3$:   // Restart to stabilize
     • $X \gets Q_2 X$
     • $R_2 \gets XX^\top$
     • $Q_2 \gets I$
   • $Z_t \gets b_t R_{t-1} + c_t R_{t-1}^2$
   • $Q_t \gets Q_{t-1} Z_t + a_t Q_{t-1}$
   • $RZ_t \gets R_{t-1} Z_t + a_t R_{t-1}$
   • $R_t \gets Z_t (RZ_t) + a_t (RZ_t)$
7. $X \gets Q_4 X$
8. If $m < n$:   $X \gets X^\top$   // Undo trick
9. Return $X$

Installation

Requirements:

• NVIDIA Hopper (H100) or Blackwell (B200/B300) GPU
• PyTorch 2.7.1+
• CUDA 12.9+

Install PyTorch first, then install from PyPI with pip install gram-newton-schulz --no-build-isolation or from source:

pip install . --no-build-isolation

--no-build-isolation is required so that pip uses your existing CUDA-enabled PyTorch instead of installing torch-cpu in an isolated build environment.

This will install:

• gram-newton-schulz (this package)
• nvidia-cutlass-dsl 4.5.2
• quack-kernels 0.5.0

Usage

WARNING: torch.compile is known to sometimes have issues with Blackwell, TORCH_COMPILE_DISABLE=1 to disable torch.compile.

Gram Newton-Schulz

from gram_newton_schulz import GramNewtonSchulz, POLAR_EXPRESS_COEFFICIENTS

gram_NS = GramNewtonSchulz(
            ns_coefficients=POLAR_EXPRESS_COEFFICIENTS,
            gram_newton_schulz_reset_iterations=[2]
        )
result = gram_NS(X)

GramNewtonSchulz is a callable function that is initialized with ns_coefficients (List of List of floats) and a list of gram_newton_schulz_reset_iterations immediately after which to restart Gram Newton-Schulz's iterative loop (List of ints) for stability. For example, [2] means a restart occurs after the 2nd iteration and [2,4] means a restart occurs after the 2nd iteration and then after the 4th.

To find the best num-restarts restart location(s) for a set of coefficients, run

python -m gram_newton_schulz.autotune_restarts --num-restarts 1 --coefs "4.0848,-6.8946,2.9270;3.9505,-6.3029,2.6377;3.7418,-5.5913,2.3037;2.8769,-3.1427,1.2046;2.8366,-3.0525,1.2012"

For 5 steps of Newton-Schulz, we recommend num-restarts = 1 for maximum speed while maintaining numerical stability. However, users who experience numerical instability or use more than 5 steps should consider using more restarts.

Muon

The Muon class supports an auxiliary scalar optimizer that updates all non-Muon parameters, custom functions that split model weights for orthogonalization, and Gram Newton-Schulz with autotuned restart locations.

import torch
from torch.optim import AdamW
from gram_newton_schulz import Muon, YOU_COEFFICIENTS

qkv_params = []
regular_2d_params = []
scalar_params = []

for name, param in model.named_parameters():
    if 'qkv_weight' in name:
      qkv_params.append(param)
    elif param.ndim >= 2:
      regular_2d_params.append(param)
    else:
      scalar_params.append(param)

scalar_optimizer = AdamW(
    scalar_params,
    lr=1e-3,
    betas=(0.9, 0.95),
    weight_decay=0.1,
)

def qkv_split_fn(param: torch.Tensor):
    """
    Split Wqkv into [Wq, Wk, Wv].

    Assumes param has shape (3*hidden_dim, hidden_dim) where the first dimension
    is concatenated [Q, K, V] weights.
    """
    hidden_dim = param.size(1)
    Wq = param[:hidden_dim, :]
    Wk = param[hidden_dim:2*hidden_dim, :]
    Wv = param[2*hidden_dim:, :]
    return [Wq, Wk, Wv]

def qkv_recombine_fn(splits):
    """Recombine [Wq, Wk, Wv] back into Wqkv."""
    return torch.cat(splits, dim=0)

muon_param_groups = []

muon_param_groups.append({
    'params': qkv_params,
    'param_split_fn': qkv_split_fn,
    'param_recombine_fn': qkv_recombine_fn,
    'lr': 3e-3,
    'weight_decay': 0.1,
    'momentum': 0.95,
})

muon_param_groups.append({
    'params': regular_2d_params,
    'lr': 3e-3,
    'weight_decay': 0.1,
    'momentum': 0.95,
})

optimizer = Muon(
    params=muon_param_groups,
    scalar_optimizer=scalar_optimizer,
    lr=3e-3,
    weight_decay=0.1,
    momentum=0.95,
    nesterov=True,
    adjust_lr='rms_norm',
    ns_algorithm='gram_newton_schulz',
    ns_use_kernels=True,
    ns_coefficients=YOU_COEFFICIENTS,
    gram_newton_schulz_num_restarts=1,
)

See example.py for a full training example.

Citation

If you use this codebase, or otherwise find our work valuable, please cite Gram Newton-Schulz:

@misc{GramNewtonSchulz,
  title   = {Gram Newton-Schulz},
  author  = {Jack Zhang and Noah Amsel and Berlin Chen and Tri Dao},
  year    = {2026},
  url     = {https://dao-ailab.github.io/blog/2026/gram-newton-schulz/}
}