build/torch210-cxx11-cu126-x86_64-linux/newton_schulz.py

from itertools import repeat
from math import inf, sqrt

import numpy as np
import torch

from .matmul_transpose_triton import matmul_transpose_assign

COMM_DTYPE = torch.bfloat16
DEFAULT_CHUNK_SIZE_RATIO = 4


def _optimal_quintic(l, u, max_iter=1000):
    """
    Use the simplified Remez algorithm to find the optimal odd quintic approximant
    to the constant function x -> 1 over the interval [l, u].

    Returns (a, b, c) for p(x) = ax + bx^3 + cx^5 that minimizes the maximum
    approximation error max_{x in [l,u]} |p(x) - 1|. Iterates by updating the
    two interior equioscillation nodes q, r until convergence. Returns the
    closed-form equioscillating solution when l ≈ u.

    Raises ValueError if any intermediate value (a, b, c, E, q, r) is non-finite
    (NaN or inf). Raises RuntimeError if convergence is not reached within
    max_iter iterations.
    """
    assert 0 <= l <= u
    if 1 - 5e-6 <= l / u:
        return (15 / 8) / u, (-10 / 8) / (u**3), (3 / 8) / (u**5)
    q = (3 * l + u) / 4
    r = (l + 3 * u) / 4
    E = inf
    for _ in range(max_iter):
        old_E = E
        LHS = np.array(
            [
                [l, l**3, l**5, 1],
                [q, q**3, q**5, -1],
                [r, r**3, r**5, 1],
                [u, u**3, u**5, -1],
            ]
        )
        a, b, c, E = np.linalg.solve(LHS, np.ones(4))
        if not np.all(np.isfinite([a, b, c, E])):
            raise ValueError(
                f"_optimal_quintic: non-finite solve result a={a}, b={b}, c={c}, E={E}"
            )
        q, r = np.sqrt(
            (-3 * b + np.array([-1, 1]) * sqrt(9 * b**2 - 20 * a * c)) / (10 * c)
        )
        if not np.all(np.isfinite([q, r])):
            raise ValueError(f"_optimal_quintic: non-finite node update q={q}, r={r}")
        if abs(old_E - E) <= 1e-15:
            break
    else:
        raise RuntimeError(
            f"_optimal_quintic: did not converge after {max_iter} iterations"
        )
    return float(a), float(b), float(c)


def _optimal_composition(l, num_iters, safety_factor_eps=0, cushion=0):
    """
    Compute the Polar Express coefficient series for `num_iters` quintic iterations.

    Builds a sequence of per-step optimal odd quintic coefficients (a, b, c) that
    compose to map singular values from [l, 1] toward 1. At each step:
      1. Solves `_optimal_quintic` on [max(l, cushion*u), u]. The `cushion`
         prevents near-zero singular values from stalling by raising the effective
         lower bound; if it is active (cushion*u > l), the coefficients are
         rescaled so that p(l) and p(u) are centered around 1 w.r.t. the true [l, u].
      2. Deflates the coefficients by (1 + safety_factor_eps)^degree for all but the
         last iteration, providing numerical headroom at the cost of a slightly slower
         final convergence step.
      3. Advances the interval: l <- p(l), u <- 2 - p(l) (by symmetry of p around 1).

    Returns a list of (a, b, c) tuples, one per iteration.

    Reference: Amsel et al., "The Polar Express: Optimal Matrix Sign Methods and
    Their Application to the Muon Algorithm", https://arxiv.org/abs/2505.16932
    """
    u = 1
    assert 0 <= l <= u
    safety_factor = 1 + safety_factor_eps
    coefficients = []
    for iter in range(num_iters):
        a, b, c = _optimal_quintic(max(l, cushion * u), u)
        if cushion * u > l:
            pl = a * l + b * l**3 + c * l**5
            pu = a * u + b * u**3 + c * u**5
            rescaler = 2 / (pl + pu)
            a *= rescaler
            b *= rescaler
            c *= rescaler
        if iter < num_iters - 1:
            a /= safety_factor
            b /= safety_factor**3
            c /= safety_factor**5
        coefficients.append((a, b, c))
        l = a * l + b * l**3 + c * l**5
        u = 2 - l
    return coefficients


# Precomputed Polar Express coefficients (a, b, c) for 10 quintic Newton-Schulz
# iterations. Each tuple is the minimax-optimal (Remez/equioscillation) odd quintic
# approximant to x->1 over the current singular-value interval, computed once at
# import time and reused across all optimizer steps.
#
# Contrast with the former hardcoded NS coefficients (5 fixed tuples):
#   - Former: empirically tuned to maximize slope at zero; did not converge
#     singular values to 1, yielding US'V^T with S' ~ Uniform(0.5, 1.5) instead
#     of the true polar factor UV^T.
#   - Polar Express: analytically optimal per step, adapting to the shrinking
#     singular-value interval [l, u] as iterations progress; converges all
#     singular values to 1, producing the exact polar factor UV^T.
_coeffs_list = _optimal_composition(
    l=1e-3, num_iters=10, safety_factor_eps=1e-2, cushion=0.02
)


# This code is adapted from:
#   KellerJordan/Muon (https://github.com/KellerJordan/Muon/blob/master/muon.py)
#   NoahAmsel/PolarExpress (https://github.com/NoahAmsel/PolarExpress)
#   matmul_transpose_assign kernel from nil0x9/flash-muon (https://github.com/nil0x9/flash-muon)
@torch.no_grad()
def _zeropower_via_newtonschulz5(G, steps):
    """
    Compute the polar factor of G via the Polar Express method.

    Applies `steps` quintic iterations X <- aX + bX^3 + cX^5, where (a, b, c)
    are the Polar Express coefficients from `_coeffs_list`. Each step is the
    optimal odd quintic approximant to x -> 1 over the current singular-value
    interval, minimizing the maximum approximation error (Remez / minimax criterion).
    The composition maps singular values from [l, 1] to near 1, producing the
    polar factor (orthogonal factor in the polar decomposition G = UP).

    `_coeffs_list` is precomputed for 10 iterations (l=1e-3, safety_factor_eps=1e-2,
    cushion=0.02). If `steps` exceeds 10, the final coefficient set is repeated.

    Reference: Amsel et al., "The Polar Express: Optimal Matrix Sign Methods and
    Their Application to the Muon Algorithm", https://arxiv.org/abs/2505.16932
    """
    assert len(G.shape) == 2
    assert G.dtype == COMM_DTYPE
    X = G  # no manual typecast

    if G.size(0) > G.size(1):
        X = X.T

    X = X / (X.norm() + 1e-7)
    hs = _coeffs_list[:steps] + list(
        repeat(_coeffs_list[-1], steps - len(_coeffs_list))
    )
    buf1 = torch.empty(X.size(0), X.size(0), dtype=X.dtype, device=X.device)
    buf2 = torch.empty(X.size(0), X.size(0), dtype=X.dtype, device=X.device)
    # Perform the NS iterations
    for a, b, c in hs:
        matmul_transpose_assign(X, buf1)
        matmul_transpose_assign(buf1, buf2)
        buf1.mul_(b).add_(buf2, alpha=c)
        X = torch.addmm(X, buf1, X, alpha=1.0, beta=a)

    if G.size(0) > G.size(1):
        X = X.T

    return X


@torch.no_grad()
def _zeropower_via_newtonschulz5_batched(G, steps):
    """Batched polar factor computation for 3D (E, out, in) tensors.

    Same algorithm as ``_zeropower_via_newtonschulz5`` but uses
    ``torch.bmm`` / ``torch.baddbmm`` instead of the 2D Triton kernel,
    processing all E expert matrices in a single batched call.
    """
    assert len(G.shape) == 3
    assert G.dtype == COMM_DTYPE
    X = G

    if G.size(1) > G.size(2):
        X = X.transpose(-2, -1)

    # Per-expert Frobenius norm.
    X = X / (X.norm(dim=(-2, -1), keepdim=True) + 1e-7)

    hs = _coeffs_list[:steps] + list(
        repeat(_coeffs_list[-1], steps - len(_coeffs_list))
    )
    for a, b, c in hs:
        buf1 = torch.bmm(X, X.transpose(-2, -1))
        buf2 = torch.bmm(buf1, buf1.transpose(-2, -1))
        buf1.mul_(b).add_(buf2, alpha=c)
        X = torch.baddbmm(X, buf1, X, alpha=1.0, beta=a)

    if G.size(1) > G.size(2):
        X = X.transpose(-2, -1)

    return X


_ns_per_shape: dict[tuple[int, ...], callable] = {}
_use_compile = True


def set_ns_compile(enabled: bool):
    """Toggle torch.compile for Newton-Schulz iteration."""
    global _use_compile
    _use_compile = enabled


def zeropower_via_newtonschulz5(G, steps=5):
    if not _use_compile:
        return _zeropower_via_newtonschulz5(G, steps)
    key = G.shape
    if key not in _ns_per_shape:
        _ns_per_shape[key] = torch.compile(_zeropower_via_newtonschulz5,
                                           options={
                                               "triton.cudagraphs": True,
                                               "shape_padding": False
                                           })
    torch.compiler.cudagraph_mark_step_begin()
    return _ns_per_shape[key](G, steps).clone()


def zeropower_via_newtonschulz5_batched(G, steps=5):
    """Compile-cached batched Newton-Schulz for 3D expert tensors."""
    if not _use_compile:
        return _zeropower_via_newtonschulz5_batched(G, steps)
    key = G.shape
    if key not in _ns_per_shape:
        _ns_per_shape[key] = torch.compile(
            _zeropower_via_newtonschulz5_batched,
            options={
                "triton.cudagraphs": True,
                "shape_padding": False
            })
    torch.compiler.cudagraph_mark_step_begin()
    return _ns_per_shape[key](G, steps).clone()