Add Newton Schultz via Polar Express as a retraction for Iso optimizer (#294)

* Add Newton Schultz via Polar Express as a retraction for Iso optimizer

Signed-off-by: Konstantin Golobokov <kogolobo@uw.edu>

Author: kogolobo

benchmarks/benchmark_stiefel_retraction.py

@@ -0,0 +1,185 @@
+# 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.
+"""Benchmark Stiefel retractions in Iso: latency, throughput, and numerical precision."""
+
+from collections.abc import Callable
+from typing import Any
+
+import torch
+import triton
+
+from emerging_optimizers import utils
+from emerging_optimizers.riemannian_optimizers.retractions.stiefel import (
+    cayley_retraction,
+    newton_schulz_retraction,
+    polar_retraction,
+    qr_retraction,
+)
+
+
+def bench(fn: Callable[[], Any], warmup: int = 25, rep: int = 100) -> float:
+    """Time fn with triton.testing.do_bench (returns execution time in ms)."""
+    return triton.testing.do_bench(fn, warmup=warmup, rep=rep)
+
+
+def test_numerical_accuracy(
+    ref: torch.Tensor,
+    qr: torch.Tensor,
+    cayley: torch.Tensor,
+    ns5: torch.Tensor,
+    ns8: torch.Tensor,
+) -> dict[str, float]:
+    """Verify factor orthogonality and error vs exact polar factor in a single batched D2H sync."""
+    k = ref.shape[-1]
+    eye = torch.eye(k, device=ref.device, dtype=ref.dtype)
+
+    # Orthogonality deviations on GPU: ||Q^T Q - I||_max
+    orth_polar = (ref.mT @ ref - eye).abs().max()
+    orth_qr = (qr.mT @ qr - eye).abs().max()
+    orth_cayley = (cayley.mT @ cayley - eye).abs().max()
+    orth_ns5 = (ns5.mT @ ns5 - eye).abs().max()
+    orth_ns8 = (ns8.mT @ ns8 - eye).abs().max()
+
+    # Approximation error vs analytical polar factor on GPU
+    ref_f = ref.float()
+    diff_ns5 = (ns5.float() - ref_f).abs()
+    err_ns5_abs = diff_ns5.max()
+    err_ns5_rel = (diff_ns5 / ref_f.abs().clamp_min(1e-8)).max()
+
+    diff_ns8 = (ns8.float() - ref_f).abs()
+    err_ns8_abs = diff_ns8.max()
+    err_ns8_rel = (diff_ns8 / ref_f.abs().clamp_min(1e-8)).max()
+
+    # Batch all 9 metrics into a single device-to-host synchronization
+    metrics = torch.stack(
+        [
+            orth_polar,
+            orth_qr,
+            orth_cayley,
+            orth_ns5,
+            orth_ns8,
+            err_ns5_abs,
+            err_ns5_rel,
+            err_ns8_abs,
+            err_ns8_rel,
+        ]
+    ).tolist()
+
+    return {
+        "orth_polar": metrics[0],
+        "orth_qr": metrics[1],
+        "orth_cayley": metrics[2],
+        "orth_ns5": metrics[3],
+        "orth_ns8": metrics[4],
+        "err_ns5_abs": metrics[5],
+        "err_ns5_rel": metrics[6],
+        "err_ns8_abs": metrics[7],
+        "err_ns8_rel": metrics[8],
+    }
+
+
+def run_case(
+    m: int,
+    n: int,
+    lr: float = 1e-3,
+    device: str = "cuda",
+    fp32_prec: utils.FP32MatmulPrecT = "high",
+) -> None:
+    """Benchmark retractions for a single parameter tensor of shape (M, N)."""
+    torch.manual_seed(0)
+    k = min(m, n)
+
+    # Initialize factor matrices U (M, K) and V (N, K) on the Stiefel manifold
+    raw_u = torch.randn(m, k, device=device, dtype=torch.float32)
+    raw_v = torch.randn(n, k, device=device, dtype=torch.float32)
+    u_init, _ = torch.linalg.qr(raw_u, mode="reduced")
+    v_init, _ = torch.linalg.qr(raw_v, mode="reduced")
+
+    # Synthetic momentum updates
+    mom_u = torch.randn_like(u_init)
+    mom_v = torch.randn_like(v_init)
+
+    # --- Correctness & Numerical Precision ---
+    ref_u = polar_retraction(u_init, mom_u, lr)
+
+    qr_u = qr_retraction(u_init, mom_u, lr)
+    cayley_u = cayley_retraction(u_init, mom_u, lr)
+    ns5_u = newton_schulz_retraction(u_init, mom_u, lr, num_ns_steps=5)
+    ns8_u = newton_schulz_retraction(u_init, mom_u, lr, num_ns_steps=8)
+
+    acc = test_numerical_accuracy(ref_u, qr_u, cayley_u, ns5_u, ns8_u)
+
+    # --- Latency Benchmarking (Factor U + Factor V) ---
+    with utils.fp32_matmul_precision(fp32_prec):
+        t_polar = bench(lambda: (polar_retraction(u_init, mom_u, lr), polar_retraction(v_init, mom_v, lr)))
+        t_qr = bench(lambda: (qr_retraction(u_init, mom_u, lr), qr_retraction(v_init, mom_v, lr)))
+        t_cayley = bench(
+            lambda: (
+                cayley_retraction(u_init, mom_u, lr),
+                cayley_retraction(v_init, mom_v, lr),
+            )
+        )
+        t_ns5 = bench(
+            lambda: (
+                newton_schulz_retraction(u_init, mom_u, lr, num_ns_steps=5),
+                newton_schulz_retraction(v_init, mom_v, lr, num_ns_steps=5),
+            )
+        )
+        t_ns8 = bench(
+            lambda: (
+                newton_schulz_retraction(u_init, mom_u, lr, num_ns_steps=8),
+                newton_schulz_retraction(v_init, mom_v, lr, num_ns_steps=8),
+            )
+        )
+
+    tag = f"M={m:<5d} N={n:<5d} K={k:<5d}"
+    speedup_ns8_vs_polar = t_polar / t_ns8
+
+    print(
+        f"{tag} | polar(svd) {t_polar:8.3f} ms | qr {t_qr:7.3f} ms | cayley {t_cayley:7.3f} ms | "
+        f"ns(steps=5) {t_ns5:7.3f} ms | ns(steps=8) {t_ns8:7.3f} ms | "
+        f"ns8 speedup vs svd: {speedup_ns8_vs_polar:6.2f}x"
+    )
+    print(
+        f"{'':>4}orth drift ||Q^T Q - I||: svd={acc['orth_polar']:.2e} | qr={acc['orth_qr']:.2e} | "
+        f"cayley={acc['orth_cayley']:.2e} | ns5={acc['orth_ns5']:.2e} | ns8={acc['orth_ns8']:.2e}"
+    )
+    print(
+        f"{'':>4}ns vs polar factor: ns5 abs={acc['err_ns5_abs']:.2e} rel={acc['err_ns5_rel']:.2e} | "
+        f"ns8 abs={acc['err_ns8_abs']:.2e} rel={acc['err_ns8_rel']:.2e}\n"
+    )
+
+
+def main() -> None:
+    """Run benchmark across square and rectangular transformer projection dimensions."""
+    torch.cuda.init()
+    device_name = torch.cuda.get_device_name(0)
+    print(f"Device: {device_name}")
+
+    cases = [
+        (1024, 1024),
+        (2048, 2048),
+        (4096, 4096),
+        (4096, 2048),
+        (8192, 2048),
+    ]
+
+    print("=== Iso Stiefel Retraction Benchmark Suite ===")
+    for m, n in cases:
+        run_case(m, n)
+
+
+if __name__ == "__main__":
+    main()

emerging_optimizers/riemannian_optimizers/isospectral.py

@@ -12,7 +12,15 @@
 # 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.
-from typing import TYPE_CHECKING, Callable, Literal, override
+from typing import TYPE_CHECKING, Callable, override
+
+from emerging_optimizers.riemannian_optimizers.retractions.stiefel import (
+    RetractionT,
+    cayley_retraction,
+    newton_schulz_retraction,
+    polar_retraction,
+    qr_retraction,
+)
 
 
 if TYPE_CHECKING:
@@ -26,43 +34,6 @@
 
 __all__ = ["Iso"]
 
-RetractionT = Literal["qr", "polar", "cayley"]
-
-
-def _qr_retraction(
-    point: torch.Tensor,
-    momentum: torch.Tensor,
-    step_size: float,
-) -> torch.Tensor:
-    matrix = point - step_size * momentum
-    q, r = torch.linalg.qr(matrix, mode="reduced")
-    signs = torch.diagonal(r).sign()
-    signs.masked_fill_(signs == 0, 1)
-    return q * signs
-
-
-def _polar_retraction(
-    point: torch.Tensor,
-    momentum: torch.Tensor,
-    step_size: float,
-) -> torch.Tensor:
-    matrix = point - step_size * momentum
-    u, _, vh = torch.linalg.svd(matrix, full_matrices=False)
-    return u @ vh
-
-
-def _cayley_retraction(
-    point: torch.Tensor,
-    momentum: torch.Tensor,
-    step_size: float,
-) -> torch.Tensor:
-    direction = -momentum
-    skew = direction @ point.mT - point @ direction.mT
-    identity = torch.eye(point.shape[0], dtype=point.dtype, device=point.device)
-    lhs = identity - 0.5 * step_size * skew
-    rhs = (identity + 0.5 * step_size * skew) @ point
-    return torch.linalg.solve(lhs, rhs)
-
 
 def _retract_factors(
     u: torch.Tensor,
@@ -89,11 +60,13 @@ def _retract_factors(
         ValueError: If the retraction method is unsupported.
     """
     if retraction == "qr":
-        retract = _qr_retraction
+        retract = qr_retraction
     elif retraction == "polar":
-        retract = _polar_retraction
+        retract = polar_retraction
     elif retraction == "cayley":
-        retract = _cayley_retraction
+        retract = cayley_retraction
+    elif retraction == "newton_schulz":
+        retract = newton_schulz_retraction
     else:
         raise ValueError(f"Invalid retraction: {retraction}")
 
@@ -138,7 +111,7 @@ def __init__(
             raise ValueError(f"Invalid learning rate: {lr}")
         if not 0.0 <= momentum < 1.0:
             raise ValueError(f"Invalid momentum value: {momentum}")
-        if retraction not in ("qr", "polar", "cayley"):
+        if retraction not in ("qr", "polar", "cayley", "newton_schulz"):
             raise ValueError(f"Invalid retraction: {retraction}")
 
         defaults = {

emerging_optimizers/riemannian_optimizers/retractions/__init__.py

@@ -0,0 +1,15 @@
+# SPDX-FileCopyrightText: Copyright (c) 2025 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.
+from emerging_optimizers.riemannian_optimizers.retractions.stiefel import *