From 1341ffe51690b6e2248cce33a2177a5ee6126ad5 Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Mon, 12 Oct 2026 12:32:03 -0700 Subject: [PATCH 1/9] add primer Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 231 ++++++++++++++++++++++++++ docs/assets/newton-schulz-quality.png | Bin 0 -> 269274 bytes docs/primer/index.md | 1 + docs/primer/newton-schulz-quality.md | 26 +++ 4 files changed, 258 insertions(+) create mode 100644 benchmarks/newton_schulz_quality.py create mode 100644 docs/assets/newton-schulz-quality.png create mode 100644 docs/primer/newton-schulz-quality.md diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py new file mode 100644 index 00000000..94782abc --- /dev/null +++ b/benchmarks/newton_schulz_quality.py @@ -0,0 +1,231 @@ +# SPDX-FileCopyrightText: Copyright (c) 2026 NVIDIA CORPORATION & AFFILIATES. All rights reserved. +# SPDX-License-Identifier: Apache-2.0 + +"""Evaluate Newton-Schulz quality on synthetic Kimi-K3 routed-expert matrices. + +Run from the repository root with ``python -m benchmarks.newton_schulz_quality``. +See docs/primer/newton-schulz-quality.md for the experiment and interpretation. +""" + +import json +import math +from pathlib import Path +from typing import Any + +import torch +from absl import app, flags + +from emerging_optimizers.orthogonalized_optimizers.muon_utils import _COEFFICIENT_SETS, newton_schulz +from emerging_optimizers.utils import fp32_matmul_precision + + +# Logical, unpacked (out_features, in_features), not the MXFP4 storage shapes. +# https://huggingface.co/moonshotai/Kimi-K3/blob/main/config.json +# Transposing the same input preserves the quality metrics; evaluate one orientation. +EXPERT_SHAPES = {"gate_up": (3072, 3584)} +FLAGS = flags.FLAGS + + +def make_matrix(shape: tuple[int, int], spectrum: str, seed: int, device: str) -> torch.Tensor: + """Create an FP32 Gaussian, uniform, or rectangular Hilbert matrix.""" + generator = torch.Generator(device=device).manual_seed(seed) + if spectrum == "gaussian": + return torch.randn(shape, generator=generator, device=device, dtype=torch.float32) + if spectrum == "uniform": + return torch.rand(shape, generator=generator, device=device, dtype=torch.float32) + if spectrum == "hilbert": + rows = torch.arange(shape[0], device=device, dtype=torch.float32)[:, None] + columns = torch.arange(shape[1], device=device, dtype=torch.float32)[None, :] + return 1.0 / (rows + columns + 1.0) + raise ValueError(f"Unknown input distribution: {spectrum}") + + +def polar_reference(x: torch.Tensor) -> tuple[torch.Tensor, float]: + """Compute the FP64 polar factor of the same FP32 input and its residual.""" + if not hasattr(torch.linalg, "polar"): + raise RuntimeError("This experiment requires torch.linalg.polar (PyTorch 2.14 or newer).") + # torch.linalg.polar requires rows >= columns. The wide polar factor follows + # by transposing the tall input's factor back; H belongs to the tall problem. + wide = x.shape[0] < x.shape[1] + tall = (x.mT if wide else x).double() + q, h = torch.linalg.polar(tall) + residual = (torch.linalg.vector_norm(q @ h - tall) / torch.linalg.vector_norm(tall)).item() + return (q.mT if wide else q), residual + + +def quality_metrics(x: torch.Tensor, reference: torch.Tensor) -> dict[str, float]: + """Measure polar error, semi-orthogonality, direction, and scale in FP64.""" + x = x.double() + reference = reference.double() + rank = min(x.shape) + gram = x @ x.mT if x.shape[0] <= x.shape[1] else x.mT @ x + identity = torch.eye(rank, device=x.device, dtype=x.dtype) + norm = torch.linalg.vector_norm(x) + reference_norm = torch.linalg.vector_norm(reference) + return { + "relative_polar_error": (torch.linalg.vector_norm(x - reference) / reference_norm).item(), + "orthogonality_error": (torch.linalg.vector_norm(gram - identity) / math.sqrt(rank)).item(), + "cosine_similarity": ((x * reference).sum() / (norm * reference_norm)).item(), + "rms_ratio": (norm / reference_norm).item(), + } + + +def plot_results(rows: list[dict[str, Any]], path: Path) -> None: + """Plot each matrix and spectrum separately, with precision encoded by line style.""" + import matplotlib + + matplotlib.use("Agg") + import matplotlib.pyplot as plt + + cases = list(dict.fromkeys((row["weight"], row["spectrum"]) for row in rows)) + fig, axes = plt.subplots(len(cases), 2, figsize=(13, 3.4 * len(cases)), squeeze=False) + colors = dict(zip(_COEFFICIENT_SETS, plt.get_cmap("tab10").colors)) + styles = {"highest": "-", "high": ":", "medium": "--"} + for axes_row, (weight, spectrum) in zip(axes, cases): + selected = [row for row in rows if (row["weight"], row["spectrum"]) == (weight, spectrum)] + for ax, metric in zip(axes_row, ("relative_polar_error", "orthogonality_error")): + for coefficient in _COEFFICIENT_SETS: + if coefficient in {"simple", "aol", "cans"}: + continue + for precision in dict.fromkeys(row["precision"] for row in selected): + group = [ + row for row in selected if row["coefficient"] == coefficient and row["precision"] == precision + ] + if not group: + continue + ax.semilogy( + [row["steps"] for row in group], + [row[metric] for row in group], + color=colors[coefficient], + linestyle=styles[precision], + marker=".", + label=f"{coefficient} / {precision}", + ) + ax.set(title=f"{weight} {selected[0]['shape']} / {spectrum}", xlabel="NS steps", ylabel=metric) + ax.set_xticks(sorted({row["steps"] for row in selected})) + ax.grid(False, which="both") + handles, labels = axes[0, 0].get_legend_handles_labels() + fig.legend(handles, labels, loc="upper center", ncol=4, fontsize=8, frameon=False) + fig.tight_layout(rect=(0, 0, 1, 1 - 1.05 / fig.get_figheight())) + path.parent.mkdir(parents=True, exist_ok=True) + fig.savefig(path, dpi=160, transparent=True) + plt.close(fig) + + +@torch.no_grad() +def main(_: list[str]) -> None: + """Sweep repository coefficient schedules and save reproducible quality measurements.""" + if not hasattr(torch.linalg, "polar"): + raise RuntimeError("This experiment requires torch.linalg.polar (PyTorch 2.14 or newer).") + steps = sorted(set(int(value) for value in FLAGS.steps)) + if not steps or min(steps) < 1: + raise ValueError("--steps must contain positive integers.") + if FLAGS.shape_divisor < 1 or any( + size % FLAGS.shape_divisor for shape in EXPERT_SHAPES.values() for size in shape + ): + raise ValueError("--shape_divisor must be positive and divide both expert dimensions.") + if not FLAGS.coefficients: + raise ValueError("--coefficients must contain at least one coefficient set.") + for coefficient in FLAGS.coefficients: + if coefficient not in _COEFFICIENT_SETS: + raise ValueError(f"Unknown coefficient set: {coefficient}") + if "cubic5" in FLAGS.coefficients and 5 not in steps: + raise ValueError("Include 5 in --steps when requesting the fixed cubic5 schedule.") + if not FLAGS.precisions or any(precision not in ("highest", "high", "medium") for precision in FLAGS.precisions): + raise ValueError("--precisions must contain highest, high, or medium.") + if not FLAGS.spectra or any(spectrum not in ("gaussian", "uniform", "hilbert") for spectrum in FLAGS.spectra): + raise ValueError("--spectra must contain gaussian, uniform, or hilbert.") + + torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False + + metadata = { + "torch_version": torch.__version__, + "cuda_version": torch.version.cuda, + "device": FLAGS.device, + "device_name": torch.cuda.get_device_name() if FLAGS.device == "cuda" else "cpu", + "seed": FLAGS.seed, + "shape_divisor": FLAGS.shape_divisor, + "input_dtype": "float32", + "reference": "torch.linalg.polar", + "reference_dtype": "float64", + "metric_dtype": "float64", + "steps": steps, + "precisions": FLAGS.precisions, + "input_distributions": FLAGS.spectra, + "uniform_interval": "[0, 1)", + "hilbert_definition": "H[i, j] = 1 / (i + j + 1), zero-based indices, constructed in float32", + "allow_bf16_reduced_precision_reduction": torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction, + "coefficient_sets": {key: _COEFFICIENT_SETS[key] for key in FLAGS.coefficients}, + } + print(json.dumps(metadata, indent=2), flush=True) + rows = [] + references = [] + for weight, full_shape in EXPERT_SHAPES.items(): + shape = tuple(size // FLAGS.shape_divisor for size in full_shape) + for spectrum in FLAGS.spectra: + print(f"Building {weight} {shape} / {spectrum}", flush=True) + x = make_matrix(shape, spectrum, FLAGS.seed, FLAGS.device) + reference, residual = polar_reference(x) + baseline = quality_metrics(reference, reference) + if ( + not math.isfinite(residual) + or not all(math.isfinite(value) for value in baseline.values()) + or residual > 1e-8 + or baseline["orthogonality_error"] > 1e-8 + ): + raise RuntimeError(f"Polar reference failed validation: residual={residual}, metrics={baseline}") + references.append( + { + "weight": weight, + "shape": shape, + "spectrum": spectrum, + "reconstruction_error": residual, + **baseline, + } + ) + for precision in FLAGS.precisions: + for coefficient in FLAGS.coefficients: + for count in [5] if coefficient == "cubic5" else steps: + with fp32_matmul_precision(precision): + result = newton_schulz(x, steps=count, coefficient_type=coefficient) + metrics = quality_metrics(result, reference) + if not all(math.isfinite(value) for value in metrics.values()): + raise RuntimeError( + f"Nonfinite result: {weight}/{spectrum}/{precision}/{coefficient}/{count}" + ) + row = { + "weight": weight, + "shape": shape, + "spectrum": spectrum, + "precision": precision, + "coefficient": coefficient, + "steps": count, + **metrics, + } + rows.append(row) + print( + f"{weight:7s} {spectrum:8s} {precision:7s} {coefficient:13s} {count:2d} " + f"polar={metrics['relative_polar_error']:.4e} " + f"orth={metrics['orthogonality_error']:.4e}", + flush=True, + ) + output = Path(FLAGS.output) + output.parent.mkdir(parents=True, exist_ok=True) + output.write_text(json.dumps({"metadata": metadata, "references": references, "results": rows}, indent=2) + "\n") + if FLAGS.plot: + plot_results(rows, Path(FLAGS.plot)) + + +if __name__ == "__main__": + flags.DEFINE_enum("device", "cuda", ["cpu", "cuda"], "Device for inputs, NS, and polar reference.") + flags.DEFINE_integer("seed", 0, "Random seed; identical inputs are reused across the sweep.") + flags.DEFINE_list("steps", ["5", "8", "10", "16"], "NS step counts.") + flags.DEFINE_list("coefficients", list(_COEFFICIENT_SETS), "Repository coefficient sets to evaluate.") + flags.DEFINE_list("precisions", ["highest", "medium"], "NS matmul precisions; medium explicitly uses BF16.") + flags.DEFINE_list( + "spectra", ["gaussian", "uniform", "hilbert"], "Inputs: N(0, 1), uniform [0, 1), or Hilbert H[i,j]=1/(i+j+1)." + ) + flags.DEFINE_integer("shape_divisor", 1, "Divide dimensions for smoke tests; 1 uses actual expert dimensions.") + flags.DEFINE_string("output", "/tmp/newton-schulz-quality.json", "Output JSON path.") + flags.DEFINE_string("plot", None, "Optional figure path; requires matplotlib.") + app.run(main) diff --git a/docs/assets/newton-schulz-quality.png b/docs/assets/newton-schulz-quality.png new file mode 100644 index 0000000000000000000000000000000000000000..d61c10a214a0be34621c90d24d7ea4d7fb289540 GIT binary patch literal 269274 zcmeFZ2T+rJxAq%|5)}dSD4?`pP>?Pvy@>{pAfWUPqVy7a4G`o}B1kVvw-AsH(tDK9 zn}YNfF@V%S5JITmt?%CNe)m3e=9_b7&dfgF>ieDF<75s03N|fFW>%NYQ}lNt5-A<{`n;*;Z}Q-_g@}RQwE>==d)KdbUV)8j{BGU+&4ig|MBW8 zKE0q5|MKE*6927V?05=6xC)gBjT;H}MC5!j_Dh^Z?Nz8E1mU`f`z~H)iBa=;H5HAw z&c&9O_k2Y1X(4)Jv6|22VSE~Si!=AmLC;4GJaD=Ztlsg--3sF55`|rA+f=A~Q7Ov3 z-&)+c(xsKV_CZXVty@YXA;<6{1~{*`&L9Zi=Guql?lw*j!EI(9xOrU>cKh7VwT`}L znrb`TEmD}X2cN^wsaQv1r%la$!?DJHL2oh$2E9ijT+Ge0d1R&nzWy|X)*3m>6-RGR zb33k6RP3(ZNoj zr@Y5@C7MS}uA?kr$N6c^qupoVbIVvX;OSUlbR0O{yhGjf$FFG-hPu4$xB0r$-_7GjV~&lsjDZ$6->_H#QH 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a/docs/primer/newton-schulz-quality.md b/docs/primer/newton-schulz-quality.md new file mode 100644 index 00000000..8ec14844 --- /dev/null +++ b/docs/primer/newton-schulz-quality.md @@ -0,0 +1,26 @@ +# Newton–Schulz quality + +## Run settings + +| Setting | Value | +| --- | --- | +| Matrix shape | `(3072, 3584)`. From Kimi-K3 MoE layers | +| Inputs | FP32 Gaussian `N(0, 1)`, uniform `[0, 1)`, and Hilbert | +| Steps | 5, 8, 10, 16; `cubic5` at 5 only | +| NS precision | FP32 (`highest`) and BF16 (`medium`); FP32 normalization and returned output | +| BF16 reduced-precision reduction | `torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False` | +| Reference | `torch.linalg.polar` in FP64 on the transposed input; factor transposed back | + +```bash +python -m benchmarks.newton_schulz_quality \ + --steps=5,8,10,16 --spectra=gaussian,uniform,hilbert \ + --output=/tmp/newton-schulz-quality.json \ + --plot=/tmp/newton-schulz-quality.png +``` + +## Results + +For NS output $X$ and reference $Q$, relative polar error is $\|X-Q\|_F / \|Q\|_F$; orthogonality error is $\|XX^T-I_{3072}\|_F / \sqrt{3072}$. + +![Relative polar and orthogonality errors for Gaussian, uniform, and Hilbert inputs. Solid lines: FP32. Dashed lines: BF16.](../assets/newton-schulz-quality.png) + From 42078e5008b8c46c81552d1d8e67b32c610d4b43 Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Mon, 12 Oct 2026 12:38:50 -0700 Subject: [PATCH 2/9] update format Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 3 ++- docs/assets/newton-schulz-quality.png | Bin 269274 -> 352039 bytes docs/primer/newton-schulz-quality.md | 9 ++++++++- 3 files changed, 10 insertions(+), 2 deletions(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index 94782abc..5679956e 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -104,11 +104,12 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: ax.set(title=f"{weight} {selected[0]['shape']} / {spectrum}", xlabel="NS steps", ylabel=metric) ax.set_xticks(sorted({row["steps"] for row in selected})) ax.grid(False, which="both") + ax.set_facecolor("white") handles, labels = axes[0, 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reference $Q$, relative polar error is $\|X-Q\|_F / \|Q\|_F$; orthogonality error is $\|XX^T-I_{3072}\|_F / \sqrt{3072}$. +For NS output `X` and reference `Q`: + +$$ +\begin{aligned} +\text{Relative polar error} &= \frac{\lVert X-Q\rVert_F}{\lVert Q\rVert_F} \\ +\text{Orthogonality error} &= \frac{\lVert XX^T-I_{3072}\rVert_F}{\sqrt{3072}} +\end{aligned} +$$ ![Relative polar and orthogonality errors for Gaussian, uniform, and Hilbert inputs. Solid lines: FP32. Dashed lines: BF16.](../assets/newton-schulz-quality.png) From 8aa0288e8d275a09200cfaefcb83e18c897f9f0c Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Mon, 12 Oct 2026 12:52:49 -0700 Subject: [PATCH 3/9] adjust format Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 19 ++++++++++++++++--- docs/assets/newton-schulz-quality.png | Bin 352039 -> 380341 bytes 2 files changed, 16 insertions(+), 3 deletions(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index 5679956e..7374dad2 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -78,7 +78,7 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: import matplotlib.pyplot as plt cases = list(dict.fromkeys((row["weight"], row["spectrum"]) for row in rows)) - fig, axes = plt.subplots(len(cases), 2, figsize=(13, 3.4 * len(cases)), squeeze=False) + fig, axes = plt.subplots(len(cases), 2, figsize=(13, 3.4 * len(cases) + 1), squeeze=False) colors = dict(zip(_COEFFICIENT_SETS, plt.get_cmap("tab10").colors)) styles = {"highest": "-", "high": ":", "medium": "--"} for axes_row, (weight, spectrum) in zip(axes, cases): @@ -99,6 +99,8 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: color=colors[coefficient], linestyle=styles[precision], marker=".", + linewidth=2.2, + markersize=7, label=f"{coefficient} / {precision}", ) ax.set(title=f"{weight} {selected[0]['shape']} / {spectrum}", xlabel="NS steps", ylabel=metric) @@ -106,8 +108,19 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: ax.grid(False, which="both") ax.set_facecolor("white") handles, labels = axes[0, 0].get_legend_handles_labels() - fig.legend(handles, labels, loc="upper center", ncol=4, fontsize=8, frameon=False) - fig.tight_layout(rect=(0, 0, 1, 1 - 1.05 / fig.get_figheight())) + fig.legend( + handles, + labels, + loc="upper center", + ncol=2, + fontsize=14, + frameon=False, + handlelength=3, + markerscale=1.5, + labelspacing=0.6, + columnspacing=2, + ) + fig.tight_layout(rect=(0, 0, 1, 1 - 1.85 / fig.get_figheight())) path.parent.mkdir(parents=True, exist_ok=True) fig.savefig(path, dpi=160, facecolor="white", transparent=False) plt.close(fig) diff --git a/docs/assets/newton-schulz-quality.png b/docs/assets/newton-schulz-quality.png index b239f695cf428b0c0bb5dd8624da5aa6d78eeb66..51cb9d083d9d3929beefbb3bcd561784517521a0 100644 GIT binary patch literal 380341 zcmeFZc|4cz+Ae$}(I6xZlpzf=gc6wxAq~c&gpi6dlX)tsj1`Ka%$XueMHy1&G8du9 z5TayC=6ziCThD&h^X_-Qd;hUNpZAaLvsSBBx_$5KIzd|aRi>pJODT$C z+OMXpO;LF2gmN?#xIX08Sp|f8-wpJcIeKYFR%FkJYcdAd-c*^%V)I>)) z+;tGy!Rtq{{`=RB)yv0Y{_7t`c%Jv!qx|n*$Ia*#ZT$DI1`bXhmjCj&4=t4M)qnqb zQnr}!-=C$NxJzOW&%Zxx zr*_~!$|@?~WMVuv^6@3jsswW+4Ldq+7Z=xUEpi?EYO%rQ^GjFlmvwcT54%$n24}Bb zO>s;&d?9ga$2!aA%!k9eDQkb`8+kZReXLiVi@fiNPac1>^q+9x2nY&#^U`%p6c1UM zx7TQ;wDXVUQ3w6E9JZLeP|c~|xq-<7ms z;o@3B^?!Qt?A~Ku!+_Ax+cocR@Tr#6)@l?j!9TyTNk)d#tSo4+o15FU^EM0fbCi$e 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zc52#pEP_2gb=GX`tr^W`fqj~3XV6xRU2mPFmfja!7a}pC6;EWcg~ERS5nBn6^gV1R z;E%PyRJQeprGznj-56Y+j9SqHDd~H1V=#zrW_Rad25o4$TV6~b^w9p|yQ`8)TPS-x z=v<*u#R7K6SgLJm^1FOk6j9LV(BV53nq?}g_W&%hS%e~bdMa=Z=Mpv%k-B0D19 zAdGJN?Ih??OZoCZBue|~orj~?=m@#Lf^WCF>O+Kbt3>;Mo!3(o`p?S z9O*m-mQ#)&b;QF7asoZZvdg}+z-LO8HAD+^+^r>GQSVBWRIRJjUMbhg?Cb8sf7bUO zf$whw#ZLYZVt+ly1JaQH)eHW=2*ckumidU!;Xk}=Yo**&JW&)fUq`Nn86Fp=U}T;6 E3t?W`!vFvP From 93f8388c171e058f1b86fa29ac55bdf9ce177b95 Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Mon, 12 Oct 2026 13:31:54 -0700 Subject: [PATCH 4/9] add more cases Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 51 ++++++++++++++++++++------ docs/assets/newton-schulz-quality.png | Bin 380341 -> 507088 bytes docs/primer/newton-schulz-quality.md | 10 +++-- 3 files changed, 45 insertions(+), 16 deletions(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index 7374dad2..d1242c9d 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -27,7 +27,7 @@ def make_matrix(shape: tuple[int, int], spectrum: str, seed: int, device: str) -> torch.Tensor: - """Create an FP32 Gaussian, uniform, or rectangular Hilbert matrix.""" + """Create an FP32 Gaussian, uniform, Hilbert, or dense logspace-spectrum matrix.""" generator = torch.Generator(device=device).manual_seed(seed) if spectrum == "gaussian": return torch.randn(shape, generator=generator, device=device, dtype=torch.float32) @@ -37,6 +37,18 @@ def make_matrix(shape: tuple[int, int], spectrum: str, seed: int, device: str) - rows = torch.arange(shape[0], device=device, dtype=torch.float32)[:, None] columns = torch.arange(shape[1], device=device, dtype=torch.float32)[None, :] return 1.0 / (rows + columns + 1.0) + if spectrum == "logspace": + rank = min(shape) + left, _ = torch.linalg.qr( + torch.randn(shape[0], rank, generator=generator, device=device, dtype=torch.float64), mode="reduced" + ) + right, _ = torch.linalg.qr( + torch.randn(shape[1], rank, generator=generator, device=device, dtype=torch.float64), mode="reduced" + ) + singular_values = torch.logspace(10, -30, rank, device=device, dtype=torch.float64) + # Reconstruct in FP64 before casting to the same FP32 input used by NS + # and polar. The prescribed tiny singular values are lost to rounding. + return ((left * singular_values) @ right.mT).float() raise ValueError(f"Unknown input distribution: {spectrum}") @@ -95,7 +107,7 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: continue ax.semilogy( [row["steps"] for row in group], - [row[metric] for row in group], + [row[metric] if row[metric] is not None else math.nan for row in group], color=colors[coefficient], linestyle=styles[precision], marker=".", @@ -147,8 +159,10 @@ def main(_: list[str]) -> None: raise ValueError("Include 5 in --steps when requesting the fixed cubic5 schedule.") if not FLAGS.precisions or any(precision not in ("highest", "high", "medium") for precision in FLAGS.precisions): raise ValueError("--precisions must contain highest, high, or medium.") - if not FLAGS.spectra or any(spectrum not in ("gaussian", "uniform", "hilbert") for spectrum in FLAGS.spectra): - raise ValueError("--spectra must contain gaussian, uniform, or hilbert.") + if not FLAGS.spectra or any( + spectrum not in ("gaussian", "uniform", "hilbert", "logspace") for spectrum in FLAGS.spectra + ): + raise ValueError("--spectra must contain gaussian, uniform, hilbert, or logspace.") torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False @@ -168,6 +182,8 @@ def main(_: list[str]) -> None: "input_distributions": FLAGS.spectra, "uniform_interval": "[0, 1)", "hilbert_definition": "H[i, j] = 1 / (i + j + 1), zero-based indices, constructed in float32", + "logspace_definition": "U @ diag(s) @ V.T reconstructed in float64, then cast to float32; random orthonormal U and V from Gaussian QR; s log-spaced from 1e10 to 1e-30", + "logspace_prescribed_condition_number": 1e40, "allow_bf16_reduced_precision_reduction": torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction, "coefficient_sets": {key: _COEFFICIENT_SETS[key] for key in FLAGS.coefficients}, } @@ -188,6 +204,14 @@ def main(_: list[str]) -> None: or baseline["orthogonality_error"] > 1e-8 ): raise RuntimeError(f"Polar reference failed validation: residual={residual}, metrics={baseline}") + input_spectrum = {} + if spectrum == "logspace": + actual_singular_values = torch.linalg.svdvals(x.double()) + input_spectrum = { + "input_sigma_max": actual_singular_values[0].item(), + "input_sigma_min": actual_singular_values[-1].item(), + "input_condition_number": (actual_singular_values[0] / actual_singular_values[-1]).item(), + } references.append( { "weight": weight, @@ -195,6 +219,7 @@ def main(_: list[str]) -> None: "spectrum": spectrum, "reconstruction_error": residual, **baseline, + **input_spectrum, } ) for precision in FLAGS.precisions: @@ -203,10 +228,7 @@ def main(_: list[str]) -> None: with fp32_matmul_precision(precision): result = newton_schulz(x, steps=count, coefficient_type=coefficient) metrics = quality_metrics(result, reference) - if not all(math.isfinite(value) for value in metrics.values()): - raise RuntimeError( - f"Nonfinite result: {weight}/{spectrum}/{precision}/{coefficient}/{count}" - ) + finite = all(math.isfinite(value) for value in metrics.values()) row = { "weight": weight, "shape": shape, @@ -214,18 +236,21 @@ def main(_: list[str]) -> None: "precision": precision, "coefficient": coefficient, "steps": count, - **metrics, + "status": "ok" if finite else "nonfinite", + **{key: value if math.isfinite(value) else None for key, value in metrics.items()}, } rows.append(row) print( f"{weight:7s} {spectrum:8s} {precision:7s} {coefficient:13s} {count:2d} " f"polar={metrics['relative_polar_error']:.4e} " - f"orth={metrics['orthogonality_error']:.4e}", + f"orth={metrics['orthogonality_error']:.4e} status={row['status']}", flush=True, ) output = Path(FLAGS.output) output.parent.mkdir(parents=True, exist_ok=True) - output.write_text(json.dumps({"metadata": metadata, "references": references, "results": rows}, indent=2) + "\n") + output.write_text( + json.dumps({"metadata": metadata, "references": references, "results": rows}, indent=2, allow_nan=False) + "\n" + ) if FLAGS.plot: plot_results(rows, Path(FLAGS.plot)) @@ -237,7 +262,9 @@ def main(_: list[str]) -> None: flags.DEFINE_list("coefficients", list(_COEFFICIENT_SETS), "Repository coefficient sets to evaluate.") flags.DEFINE_list("precisions", ["highest", "medium"], "NS matmul precisions; medium explicitly uses BF16.") flags.DEFINE_list( - "spectra", ["gaussian", "uniform", "hilbert"], "Inputs: N(0, 1), uniform [0, 1), or Hilbert H[i,j]=1/(i+j+1)." + "spectra", + ["gaussian", "uniform", "hilbert", "logspace"], + "Inputs: N(0, 1), uniform [0, 1), Hilbert, or dense SVD reconstruction with prescribed singular values from 1e10 to 1e-30.", ) flags.DEFINE_integer("shape_divisor", 1, "Divide dimensions for smoke tests; 1 uses actual expert dimensions.") flags.DEFINE_string("output", "/tmp/newton-schulz-quality.json", "Output JSON path.") diff --git a/docs/assets/newton-schulz-quality.png b/docs/assets/newton-schulz-quality.png 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From Kimi-K3 MoE layers | -| Inputs | FP32 Gaussian `N(0, 1)`, uniform `[0, 1)`, and Hilbert | +| Inputs | FP32 Gaussian `N(0, 1)`, uniform `[0, 1)`, Hilbert, and dense logspace | +| Logspace input | `U @ diag(s) @ V.T`; independent Gaussian QR factors `U`, `V`; `s` logarithmically spaced from `1e-30` to `1e10`; FP64 reconstruction cast to FP32 | | Steps | 5, 8, 10, 16; `cubic5` at 5 only | | NS precision | FP32 (`highest`) and BF16 (`medium`); FP32 normalization and returned output | | BF16 reduced-precision reduction | `torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False` | @@ -13,13 +14,15 @@ ```bash python -m benchmarks.newton_schulz_quality \ - --steps=5,8,10,16 --spectra=gaussian,uniform,hilbert \ + --steps=5,8,10,16 --spectra=gaussian,uniform,hilbert,logspace \ --output=/tmp/newton-schulz-quality.json \ --plot=/tmp/newton-schulz-quality.png ``` ## Results +Dense logspace input after FP32 conversion: measured singular values `1.54` to `1e+10`, condition number `6.47e+09`. + For NS output `X` and reference `Q`: $$ @@ -29,5 +32,4 @@ $$ \end{aligned} $$ -![Relative polar and orthogonality errors for Gaussian, uniform, and Hilbert inputs. Solid lines: FP32. Dashed lines: BF16.](../assets/newton-schulz-quality.png) - +![Relative polar and orthogonality errors for Gaussian, uniform, Hilbert, and dense logspace inputs. Solid lines: FP32. Dashed lines: BF16.](../assets/newton-schulz-quality.png) From 8224323845c474c56f498de385e3972c057f9ffb Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Mon, 12 Oct 2026 16:46:24 -0700 Subject: [PATCH 5/9] update measurement Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 19 ++++++++++++++++++- docs/assets/newton-schulz-quality.png | Bin 507088 -> 508545 bytes docs/primer/newton-schulz-quality.md | 1 + 3 files changed, 19 insertions(+), 1 deletion(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index d1242c9d..f761257e 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -15,7 +15,12 @@ import torch from absl import app, flags -from emerging_optimizers.orthogonalized_optimizers.muon_utils import _COEFFICIENT_SETS, newton_schulz +from emerging_optimizers.orthogonalized_optimizers.muon_utils import ( + _COEFFICIENT_ITER_MODES, + _COEFFICIENT_SETS, + get_coefficient_iterator, + newton_schulz, +) from emerging_optimizers.utils import fp32_matmul_precision @@ -186,6 +191,18 @@ def main(_: list[str]) -> None: "logspace_prescribed_condition_number": 1e40, "allow_bf16_reduced_precision_reduction": torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction, "coefficient_sets": {key: _COEFFICIENT_SETS[key] for key in FLAGS.coefficients}, + "coefficient_iteration_modes": {key: _COEFFICIENT_ITER_MODES.get(key, "cycle") for key in FLAGS.coefficients}, + "coefficient_schedules": { + key: { + str(count): list( + get_coefficient_iterator( + count, _COEFFICIENT_SETS[key], mode=_COEFFICIENT_ITER_MODES.get(key, "cycle") + ) + ) + for count in ([5] if key == "cubic5" else steps) + } + for key in FLAGS.coefficients + }, } print(json.dumps(metadata, indent=2), flush=True) rows = [] diff --git a/docs/assets/newton-schulz-quality.png b/docs/assets/newton-schulz-quality.png index 7933f25adbe1d8b494a9324c47347ee3111ef4e7..06f37a5daf80da7f0654e0d0d35f63bdec85e31d 100644 GIT binary patch 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`torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False` | | Reference | `torch.linalg.polar` in FP64 on the transposed input; factor transposed back | From 8bdeefdf61354395609108c6bfc6fb54a8c07336 Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Tue, 13 Oct 2026 10:16:49 -0700 Subject: [PATCH 6/9] change to reconstruction error Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 23 ++++++++++++++++------- docs/assets/newton-schulz-quality.png | Bin 508545 -> 522440 bytes docs/primer/newton-schulz-quality.md | 6 +++--- 3 files changed, 19 insertions(+), 10 deletions(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index f761257e..8ee7533d 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -4,7 +4,7 @@ """Evaluate Newton-Schulz quality on synthetic Kimi-K3 routed-expert matrices. Run from the repository root with ``python -m benchmarks.newton_schulz_quality``. -See docs/primer/newton-schulz-quality.md for the experiment and interpretation. +See docs/primer/newton-schulz-quality.md for the run settings and results. """ import json @@ -57,8 +57,8 @@ def make_matrix(shape: tuple[int, int], spectrum: str, seed: int, device: str) - raise ValueError(f"Unknown input distribution: {spectrum}") -def polar_reference(x: torch.Tensor) -> tuple[torch.Tensor, float]: - """Compute the FP64 polar factor of the same FP32 input and its residual.""" +def polar_reference(x: torch.Tensor) -> tuple[torch.Tensor, torch.Tensor, float]: + """Return FP64 polar factors Q, H and the relative reconstruction residual.""" if not hasattr(torch.linalg, "polar"): raise RuntimeError("This experiment requires torch.linalg.polar (PyTorch 2.14 or newer).") # torch.linalg.polar requires rows >= columns. The wide polar factor follows @@ -67,7 +67,14 @@ def polar_reference(x: torch.Tensor) -> tuple[torch.Tensor, float]: tall = (x.mT if wide else x).double() q, h = torch.linalg.polar(tall) residual = (torch.linalg.vector_norm(q @ h - tall) / torch.linalg.vector_norm(tall)).item() - return (q.mT if wide else q), residual + return (q.mT if wide else q), h, residual + + +def reconstruction_error(x: torch.Tensor, original: torch.Tensor, h: torch.Tensor) -> float: + """Reconstruct the original input with the NS output and reference polar H in FP64.""" + x, original, h = x.double(), original.double(), h.double() + reconstructed = h @ x if x.shape[0] < x.shape[1] else x @ h + return (torch.linalg.vector_norm(reconstructed - original) / torch.linalg.vector_norm(original)).item() def quality_metrics(x: torch.Tensor, reference: torch.Tensor) -> dict[str, float]: @@ -100,7 +107,7 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: styles = {"highest": "-", "high": ":", "medium": "--"} for axes_row, (weight, spectrum) in zip(axes, cases): selected = [row for row in rows if (row["weight"], row["spectrum"]) == (weight, spectrum)] - for ax, metric in zip(axes_row, ("relative_polar_error", "orthogonality_error")): + for ax, metric in zip(axes_row, ("relative_polar_error", "reconstruction_error")): for coefficient in _COEFFICIENT_SETS: if coefficient in {"simple", "aol", "cans"}: continue @@ -182,6 +189,7 @@ def main(_: list[str]) -> None: "reference": "torch.linalg.polar", "reference_dtype": "float64", "metric_dtype": "float64", + "reconstruction_error_definition": "||H_ref @ X - A||_F / ||A||_F for wide A; ||X @ H_ref - A||_F / ||A||_F otherwise; H_ref from FP64 polar of original FP32 A", "steps": steps, "precisions": FLAGS.precisions, "input_distributions": FLAGS.spectra, @@ -212,7 +220,7 @@ def main(_: list[str]) -> None: for spectrum in FLAGS.spectra: print(f"Building {weight} {shape} / {spectrum}", flush=True) x = make_matrix(shape, spectrum, FLAGS.seed, FLAGS.device) - reference, residual = polar_reference(x) + reference, h, residual = polar_reference(x) baseline = quality_metrics(reference, reference) if ( not math.isfinite(residual) @@ -245,6 +253,7 @@ def main(_: list[str]) -> None: with fp32_matmul_precision(precision): result = newton_schulz(x, steps=count, coefficient_type=coefficient) metrics = quality_metrics(result, reference) + metrics["reconstruction_error"] = reconstruction_error(result, x, h) finite = all(math.isfinite(value) for value in metrics.values()) row = { "weight": weight, @@ -260,7 +269,7 @@ def main(_: list[str]) -> None: print( f"{weight:7s} {spectrum:8s} {precision:7s} {coefficient:13s} {count:2d} " f"polar={metrics['relative_polar_error']:.4e} " - f"orth={metrics['orthogonality_error']:.4e} status={row['status']}", + f"reconstruction={metrics['reconstruction_error']:.4e} status={row['status']}", flush=True, ) output = Path(FLAGS.output) diff --git a/docs/assets/newton-schulz-quality.png 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z`!WZZ`DfVQTy_5iZhQcm`lRpl_4YIC_4B;gg-!x1tP_W5N+<$A$+tXl|joBb+rdx4M-FZ?oUQ~<&xZ@JRl|@I=Ic- z$sljH00X`Yu~%exIHL0lpGTzlkeCIG6LBE}505$oOAh=Li-05a*kfE)$1d64bn6n? znFo(iy!rd#Sd9NMf9zOsB(@2+Z%NsWey3wV?ioF%}HVg2*N>-NY8=@*AgOQL_!-$4Iu0(5}ydXq?T!EX+Z)nubtY- zmbD%0+14mzMmK@!B!3kwl$U{Cyj@Jg Date: Tue, 13 Oct 2026 10:32:17 -0700 Subject: [PATCH 7/9] minor editing Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 2 +- docs/assets/newton-schulz-quality.png | Bin 522440 -> 514884 bytes docs/primer/newton-schulz-quality.md | 4 +++- 3 files changed, 4 insertions(+), 2 deletions(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index 8ee7533d..9375cb06 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -127,7 +127,7 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: markersize=7, label=f"{coefficient} / {precision}", ) - ax.set(title=f"{weight} {selected[0]['shape']} / {spectrum}", 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benchmarks/newton_schulz_quality.py | 8 ++++++-- docs/assets/newton-schulz-quality.png | Bin 514884 -> 605559 bytes docs/primer/newton-schulz-quality.md | 5 +++-- 3 files changed, 9 insertions(+), 4 deletions(-) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index 9375cb06..7f905457 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -144,7 +144,9 @@ def plot_results(rows: list[dict[str, Any]], path: Path) -> None: labelspacing=0.6, columnspacing=2, ) - fig.tight_layout(rect=(0, 0, 1, 1 - 1.85 / fig.get_figheight())) + legend_height = 0.35 * math.ceil(len(handles) / 2) + 0.45 + fig.set_size_inches(13, 3.4 * len(cases) + legend_height - 0.85) + fig.tight_layout(rect=(0, 0, 1, 1 - legend_height / fig.get_figheight())) path.parent.mkdir(parents=True, exist_ok=True) fig.savefig(path, dpi=160, facecolor="white", transparent=False) plt.close(fig) @@ -286,7 +288,9 @@ def main(_: list[str]) -> None: flags.DEFINE_integer("seed", 0, "Random seed; identical inputs are reused across the sweep.") flags.DEFINE_list("steps", ["5", "8", "10", "16"], "NS step counts.") flags.DEFINE_list("coefficients", list(_COEFFICIENT_SETS), "Repository coefficient sets to evaluate.") - flags.DEFINE_list("precisions", ["highest", "medium"], "NS matmul precisions; medium explicitly uses BF16.") + flags.DEFINE_list( + "precisions", ["highest", "high", "medium"], "NS matmul precisions; medium explicitly uses BF16." + ) flags.DEFINE_list( "spectra", ["gaussian", "uniform", "hilbert", "logspace"], diff --git a/docs/assets/newton-schulz-quality.png b/docs/assets/newton-schulz-quality.png index 4264f7c1ec20ed34f8624ebc859c77048040cd38..a560e2d99284263ab458b3f0b5ed25125988c026 100644 GIT binary patch literal 605559 zcmeFZ2{hJw+cx}{ic%pN%aEcBk&rT_C{oExh>#4CAyhJ@6cX7PN~Ds65E;vq3`J!q zLlH$n=6Q_o__g=*e9!%?=Uwmnt?&KTyS}v^Yu|f!>*D%fzu`QO^Ei(44Aj(6UA>ZP zB}GxI52~qXQxrWPMXk6=zZ~B=(D$Al|B-dwZ|JJyc*^yhxr-Hb#N5@%-qF?G)`HL7 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`1e10`; FP64 reconstruction cast to FP32 | | Steps | 5, 8, 10, 16; `cubic5` at 5 only | | Polar Express schedule | Upstream `repeat_penultimate` coefficient iterator (main `4ebce60`) | -| NS precision | FP32 (`highest`) and BF16 (`medium`); FP32 normalization and returned output | +| NS precision | FP32 tensors with `highest` and `high` matmul precision, and BF16 (`medium`); FP32 normalization and returned output | | BF16 reduced-precision reduction | `torch.backends.cuda.matmul.allow_bf16_reduced_precision_reduction = False` | | Reference | `torch.linalg.polar` in FP64 on the transposed input; factor transposed back | ```bash python -m benchmarks.newton_schulz_quality \ + --precisions=highest,high,medium \ --steps=5,8,10,16 --spectra=gaussian,uniform,hilbert,logspace \ --output=/tmp/newton-schulz-quality.json \ --plot=/tmp/newton-schulz-quality.png @@ -35,4 +36,4 @@ $$ \end{aligned} $$ -![Relative polar and reconstruction errors for Gaussian, uniform, Hilbert, and dense logspace inputs. Solid lines: FP32. Dashed lines: BF16.](../assets/newton-schulz-quality.png) +![Relative polar and reconstruction errors for Gaussian, uniform, Hilbert, and dense logspace inputs. Solid lines: highest. Dotted lines: high. Dashed lines: medium (BF16).](../assets/newton-schulz-quality.png) From 3b734f3a046b9c604a47443aaca76a3c5fbee9ca Mon Sep 17 00:00:00 2001 From: Hao Wu Date: Tue, 13 Oct 2026 10:45:14 -0700 Subject: [PATCH 9/9] fix license header Signed-off-by: Hao Wu --- benchmarks/newton_schulz_quality.py | 12 ++++++++++++ 1 file changed, 12 insertions(+) diff --git a/benchmarks/newton_schulz_quality.py b/benchmarks/newton_schulz_quality.py index 7f905457..cbd63774 100644 --- a/benchmarks/newton_schulz_quality.py +++ b/benchmarks/newton_schulz_quality.py @@ -1,5 +1,17 @@ # SPDX-FileCopyrightText: Copyright (c) 2026 NVIDIA CORPORATION & AFFILIATES. All rights reserved. # SPDX-License-Identifier: Apache-2.0 +# +# Licensed under the Apache License, Version 2.0 (the "License"); +# you may not use this file except in compliance with the License. +# You may obtain a copy of the License at +# +# http://www.apache.org/licenses/LICENSE-2.0 +# +# Unless required by applicable law or agreed to in writing, software +# distributed under the License is distributed on an "AS IS" BASIS, +# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +# See the License for the specific language governing permissions and +# limitations under the License. """Evaluate Newton-Schulz quality on synthetic Kimi-K3 routed-expert matrices.

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