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0dc9e85 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 | #!/usr/bin/env python3
"""CPU audit of the CAffNet piecewise-constraint claim.
The paper specifies the benchmark functions and the reported Table 2 means,
but not enough transformer implementation detail to call a new training run a
faithful reproduction. This program therefore keeps two outputs separate:
* an exact Decimal audit of the published rounded means and the claimed
73.33% reduction; and
* an independent, explicitly labelled one-token, three-head transformer
experiment using the published functions and training regime.
No result is inferred from the source table: the script prints the source
arithmetic and the executed CPU experiment independently.
"""
from __future__ import annotations
import argparse
import json
import math
import random
from decimal import Decimal, getcontext
from typing import Callable
import numpy as np
import torch
from torch import nn
torch.set_num_threads(1)
torch.set_num_interop_threads(1)
getcontext().prec = 40
def pw(x: np.ndarray | torch.Tensor, *, kind: str):
"""Published Appendix D.1 functions, evaluated by the four branches."""
if isinstance(x, torch.Tensor):
sin = torch.sin
pi = torch.pi
where = torch.where
z = torch.zeros_like(x)
else:
sin = np.sin
pi = np.pi
where = np.where
z = np.zeros_like(x)
a = x <= -1
b = (x > -1) & (x <= 0)
c = (x > 0) & (x <= 1)
if kind == "f":
vals = (-5 * sin(pi / 2 * (x + 1)) - 2,
-2 + z,
2 - 9 * (x - 2 / 3) ** 2,
3 / x ** 2 - 2)
elif kind == "u1":
vals = (-3 * sin(pi / 2 * (x + 1)) + 1 / 5,
-2 + z,
3 - 4 * (x - 1 / 2) ** 2,
2 + z)
elif kind == "u2":
vals = (-3 * sin(pi / 2 * (x + 1)) ** 3 + 1,
2 + z,
3 - 4 * (x - 4 / 5) ** 2,
5 / 2 + z)
elif kind == "l1":
vals = (5 * sin(pi / 2 * (x + 1)) ** 2 - 3,
-2 + z,
(4 - 9 * (x - 2 / 3) ** 2) * x - 5 / 2,
3 / x ** 3 - 5 / 2)
elif kind == "l2":
vals = (5 * sin(pi / 2 * (x + 1)) ** 8 - 2,
-3 + z,
(5 - 4 * (x - 1 / 6) ** 2) * x - 5 / 2,
3 / (2 * x ** 3) - 16 / 9)
else:
raise ValueError(kind)
return where(a, vals[0], where(b, vals[1], where(c, vals[2], vals[3])))
def source_table_audit() -> dict[str, object]:
nn_mse = Decimal("0.0045")
tf_mse = Decimal("0.0012")
reduction = (Decimal(1) - tf_mse / nn_mse) * Decimal(100)
# Rounded four-decimal means admit an interval; report it rather than
# silently treating the displayed values as hidden unrounded results.
nn_lo, nn_hi = Decimal("0.00445"), Decimal("0.00455")
tf_lo, tf_hi = Decimal("0.00115"), Decimal("0.00125")
lo = (Decimal(1) - tf_hi / nn_lo) * Decimal(100)
hi = (Decimal(1) - tf_lo / nn_hi) * Decimal(100)
return {
"published_nn_mse": str(nn_mse),
"published_tf_mse": str(tf_mse),
"reduction_from_displayed_means_percent": str(reduction),
"reduction_interval_from_four_decimal_rounding_percent": [str(lo), str(hi)],
"displayed_zero_violation_is_not_proof_of_exact_zero": True,
}
def domain_audit() -> dict[str, float]:
x = np.linspace(-2.0, 2.0, 400_001, dtype=np.float64)
target = pw(x, kind="f")
upper = np.minimum(pw(x, kind="u1"), pw(x, kind="u2"))
lower = np.maximum(pw(x, kind="l1"), pw(x, kind="l2"))
violations = np.maximum(target - upper, 0) + np.maximum(lower - target, 0)
return {
"grid_points": float(x.size),
"target_max_constraint_residual": float(np.max(violations)),
"target_min_feasible_width": float(np.min(upper - lower)),
"target_max_feasible_width": float(np.max(upper - lower)),
}
class OneTokenTransformer(nn.Module):
"""Small explicit interpretation of 3 heads of size 40, width 120."""
def __init__(self):
super().__init__()
self.embed = nn.Linear(1, 120)
layer = nn.TransformerEncoderLayer(
d_model=120,
nhead=3,
dim_feedforward=120,
dropout=0.0,
activation="gelu",
batch_first=True,
norm_first=False,
)
self.encoder = nn.TransformerEncoder(layer, num_layers=1, enable_nested_tensor=False)
self.out = nn.Linear(120, 1)
def forward(self, x: torch.Tensor) -> torch.Tensor:
h = self.embed(x[:, None, None])
return self.out(self.encoder(h))[:, 0, 0]
def train(seed: int, epochs: int) -> dict[str, float | int]:
random.seed(seed)
np.random.seed(seed)
torch.manual_seed(seed)
dtype = torch.float64
x_train = torch.from_numpy(np.random.default_rng(seed).uniform(-2, 2, 50)).to(dtype)
y_train = pw(x_train, kind="f")
x_test = torch.linspace(-2, 2, 400, dtype=dtype)
y_test = pw(x_test, kind="f")
upper_train = torch.minimum(pw(x_train, kind="u1"), pw(x_train, kind="u2"))
lower_train = torch.maximum(pw(x_train, kind="l1"), pw(x_train, kind="l2"))
upper_test = torch.minimum(pw(x_test, kind="u1"), pw(x_test, kind="u2"))
lower_test = torch.maximum(pw(x_test, kind="l1"), pw(x_test, kind="l2"))
def fit(project: bool) -> tuple[float, float]:
model = OneTokenTransformer().to(dtype)
opt = torch.optim.Adam(model.parameters(), lr=1e-4)
for _ in range(epochs):
raw = model(x_train)
if project:
pred = torch.maximum(torch.minimum(raw, upper_train), lower_train)
loss = torch.mean((pred - y_train) ** 2)
else:
residual = torch.stack((raw - upper_train, raw - upper_train, lower_train - raw, lower_train - raw), dim=1)
# The paper specifies the 2-norm in Eq. (6), not a sum of
# row penalties. Keep the four rows explicit so the exact
# source constraint convention is visible in the run.
penalty = torch.linalg.vector_norm(torch.relu(residual), ord=2, dim=1)
loss = torch.mean((raw - y_train) ** 2 + 100 * penalty)
opt.zero_grad(set_to_none=True)
loss.backward()
opt.step()
with torch.no_grad():
raw = model(x_test)
pred = torch.maximum(torch.minimum(raw, upper_test), lower_test) if project else raw
residual = torch.stack((pred - upper_test, pred - upper_test, lower_test - pred, lower_test - pred), dim=1)
violation = torch.relu(residual)
mse = torch.mean((pred - y_test) ** 2).item()
vmax = torch.max(violation).item()
return mse, vmax
nn_mse, nn_vmax = fit(project=False)
tf_mse, tf_vmax = fit(project=True)
return {
"seed": seed,
"epochs": epochs,
"nn_mse": nn_mse,
"nn_max_violation": nn_vmax,
"caffnet_tf_mse": tf_mse,
"caffnet_tf_max_violation": tf_vmax,
"mse_reduction_percent": 100 * (nn_mse - tf_mse) / nn_mse,
}
def main() -> None:
parser = argparse.ArgumentParser()
parser.add_argument("--epochs", type=int, default=5_000)
parser.add_argument("--seeds", type=int, nargs="+", default=[0, 1, 2, 3, 4])
args = parser.parse_args()
result: dict[str, object] = {
"source_table_audit": source_table_audit(),
"domain_audit": domain_audit(),
"training": [train(seed, args.epochs) for seed in args.seeds],
}
print(json.dumps(result, indent=2, sort_keys=True))
if __name__ == "__main__":
main()
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