File size: 6,095 Bytes
1f48ccf | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | """Full-batch spherical GD on the correlation loss: overlap vs delta = n/d.
Reproduces Figures 1a/1b of arXiv:2602.02431 (paper #26332).
quad -> Theorem 3.1 (Claim 1): threshold delta grows with log d
trunc -> Theorem 3.2 (Claims 2/5): threshold delta is d-independent
"""
from __future__ import annotations
import argparse
import csv
import math
import os
import sys
import time
import torch
sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))
import sim
def a_star_chunked(X, y, chunk=16384):
n, d = X.shape
A = torch.zeros(d, d, device=X.device, dtype=X.dtype)
for i in range(0, n, chunk):
Xi = X[i : i + chunk]
A += Xi.T @ (y[i : i + chunk, None] * Xi)
return (2.0 / n) * A
def top2(A):
"""Top two eigenvalues + top eigenvector.
Full eigendecomposition is faster than LOBPCG below d ~ 3000 (LOBPCG is
kernel-launch bound at small d); above that we fall back to LOBPCG with k=2.
"""
d = A.shape[0]
if d <= 3000:
ev, evec = torch.linalg.eigh(A.double())
return float(ev[-1]), float(ev[-2]), evec[:, -1]
try:
vals, vecs = torch.lobpcg(A.double(), k=2, largest=True, niter=400, tol=1e-10)
return float(vals[0]), float(vals[1]), vecs[:, 0]
except Exception:
ev, evec = torch.linalg.eigh(A.double())
return float(ev[-1]), float(ev[-2]), evec[:, -1]
def main():
p = argparse.ArgumentParser()
p.add_argument("--act", default="trunc", choices=["quad", "trunc", "smooth"])
p.add_argument("--dims", default="64,128,256,512,1024,2048,4096")
p.add_argument("--delta-min", type=float, default=0.5)
p.add_argument("--delta-max", type=float, default=11.0)
p.add_argument("--delta-step", type=float, default=0.5)
p.add_argument("--seeds", default="32,32,32,16,16,8,8", help="per dim")
p.add_argument("--M", type=float, default=8.0)
p.add_argument("--eta", type=float, default=0.1)
p.add_argument("--T", type=int, default=3000, help="steps for non-quad activations")
p.add_argument("--spectrum", action="store_true", help="also record lam1/lam2/v1(A*)")
p.add_argument("--out", required=True)
args = p.parse_args()
dev = "cuda" if torch.cuda.is_available() else "cpu"
dims = [int(v) for v in args.dims.split(",")]
seeds = [int(v) for v in args.seeds.split(",")]
assert len(seeds) == len(dims)
deltas = [
round(args.delta_min + i * args.delta_step, 4)
for i in range(int(round((args.delta_max - args.delta_min) / args.delta_step)) + 1)
]
print(f"device={dev} act={args.act} dims={dims} seeds={seeds} deltas={deltas}", flush=True)
rows = []
t_start = time.time()
for d, ns in zip(dims, seeds):
# quadratic: A* is constant along the flow -> iterate on the d x d matrix (exact,
# and far cheaper); truncated: A(theta) is time-varying -> matrix-free matvecs.
use_matrix = args.act == "quad"
T = sim.log2_steps(d) if use_matrix else args.T
for delta in deltas:
n = int(round(delta * d))
for s in range(ns):
seed = 1000 * d + 7 * s + int(delta * 2)
t0 = time.time()
data = sim.make_data(d, n, seed, args.act, args.M, dev, torch.float32)
lam1 = lam2 = ov_v1 = float("nan")
if use_matrix or args.spectrum:
A = a_star_chunked(data.X, data.y).double()
lam1, lam2, v1 = top2(A)
ov_v1 = float((v1 @ data.theta_star.double()) ** 2)
th0 = sim.rand_sphere(d, 500_000 + seed, dev, torch.float32)
if use_matrix:
dd = sim.Data(data.X, data.y, data.theta_star.double())
theta = th0.double()
ts = dd.theta_star
prev_r = prev_o = None
steps = T
for t in range(T):
Ath = A @ theta
ray = theta @ Ath
grad = Ath - ray * theta
theta = theta + args.eta * grad
theta = theta / theta.norm()
if (t + 1) % 200 == 0:
r, o = float(ray), float((theta @ ts) ** 2)
if (
prev_r is not None
and abs(r - prev_r) <= 1e-13 * abs(r)
and abs(o - prev_o) <= 1e-13
):
steps = t + 1
break
prev_r, prev_o = r, o
ov = float((theta @ ts) ** 2)
else:
theta, steps, _ = sim.spherical_flow(
data, th0, args.act, args.M, eta=args.eta, T=T,
tol=0.0, check_every=10 ** 9,
)
ov = float((theta @ data.theta_star) ** 2)
rows.append(
dict(act=args.act, d=d, delta=delta, n=n, seed=seed, M=args.M,
eta=args.eta, T=T, steps=steps, sq_overlap=round(ov, 6),
lam1=lam1, lam2=lam2, sq_overlap_v1Astar=ov_v1,
secs=round(time.time() - t0, 3))
)
del data
if use_matrix or args.spectrum:
del A
torch.cuda.empty_cache() if dev == "cuda" else None
m = [r["sq_overlap"] for r in rows if r["d"] == d and r["delta"] == delta]
print(
f"[{time.time()-t_start:7.1f}s] d={d:5d} delta={delta:5.1f} "
f"mean ov2={sum(m)/len(m):.4f} (n={n}, {ns} seeds)",
flush=True,
)
with open(args.out, "w", newline="") as f:
w = csv.DictWriter(f, fieldnames=list(rows[0].keys()))
w.writeheader()
w.writerows(rows)
print(f"wrote {args.out} ({len(rows)} rows, {time.time()-t_start:.1f}s)")
if __name__ == "__main__":
main()
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