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No scientific training, dataset, checkpoint, or active experiment is accessed.
Two fixed token-logit scalars provide a cross-input Jacobian kernel. Actual
parameter identity determines aliases, including MLP-only sharing; the tied
input/output embedding is differentiated once with both computational paths.
Small plain-SGD steps below are finite-difference checks on disposable copies,
not claims about AdamW, generalization, or learning efficiency.
"""
from __future__ import annotations
import argparse
import copy
import hashlib
import json
import platform
import sys
from datetime import datetime, timezone
from pathlib import Path
ROOT = Path(__file__).resolve().parents[3]
sys.path.insert(0, str(ROOT / "src"))
import torch # noqa: E402
import transformers # noqa: E402
from llm_memory_editability.loop_learning import construct # noqa: E402
SEED = 20261007
TOLERANCE = 1e-10
SGD_STEPS = (1e-4, 5e-5, 2.5e-5)
SPEC = {
"initialization": SEED,
"base_unique_blocks": 1,
"repeats": 2,
"initialization_reference_layers": 8,
"model": {
"vocab_size": 11,
"positions": 16,
"hidden_size": 8,
"attention_heads": 2,
"dropout": 0.0,
},
}
OUTPUTS = {
"query": {"tokens": [[1, 4, 2]], "position": 2, "token_logit": 7},
"train": {"tokens": [[3, 1, 5]], "position": 2, "token_logit": 4},
}
def scalar_output(model, name):
setting = OUTPUTS[name]
tokens = torch.tensor(setting["tokens"], dtype=torch.long)
return model(tokens)[0, setting["position"], setting["token_logit"]]
def scalar_jacobian(model, name):
parameters = dict(model.named_parameters())
output = scalar_output(model, name)
gradients = torch.autograd.grad(output, tuple(parameters.values()))
return float(output.detach()), {
key: gradient.detach().clone() for key, gradient in zip(parameters, gradients, strict=True)
}
def dot(left, right):
return sum(float((left[key] * right[key]).sum()) for key in left)
def max_abs(value):
return float(value.abs().max())
def parameter_aliases(model, untied):
"""Map each actual unique parameter to its independent execution copies."""
unique = dict(model.named_parameters())
names = {id(parameter): name for name, parameter in unique.items()}
aliases = {name: [] for name in unique}
for name, parameter in unique.items():
if not name.startswith("transformer.h."):
aliases[names[id(parameter)]].append(name)
occurrence = 0
for _ in range(model.repeats):
for block in model.transformer.h:
for name, parameter in block.named_parameters():
aliases[names[id(parameter)]].append(f"transformer.h.{occurrence}.{name}")
occurrence += 1
independent = dict(untied.named_parameters())
flattened = [alias for group in aliases.values() for alias in group]
if len(flattened) != len(set(flattened)) or set(flattened) != set(independent):
raise AssertionError("Alias groups must partition all independent parameters")
for name, group in aliases.items():
if not group or any(not torch.equal(unique[name], independent[alias]) for alias in group):
raise AssertionError(f"Alias weights differ at common forward point: {name}")
return aliases
def check_kernel(model, untied, reference, arm):
aliases = parameter_aliases(model, untied)
scalar_values, gradients = {}, {}
gradient_errors = {}
for output in OUTPUTS:
scalar_values[output], gradients[output] = scalar_jacobian(model, output)
errors = {}
for name, group in aliases.items():
expected = torch.stack([reference[output][alias] for alias in group]).sum(0)
errors[name] = max_abs(gradients[output][name] - expected)
gradient_errors[output] = errors
common, diagonal, cross = 0.0, 0.0, 0.0
for group in aliases.values():
for left in group:
for right in group:
value = float((reference["query"][left] * reference["train"][right]).sum())
if not left.startswith("transformer.h."):
if left != right:
raise AssertionError("Common parameters must occur exactly once")
common += value
elif left == right:
diagonal += value
else:
cross += value
if arm == "mlp_shared" and not (".mlp." in left and ".mlp." in right):
raise AssertionError("MLP-only sharing introduced a non-MLP cross term")
actual = dot(gradients["query"], gradients["train"])
independent_actual = dot(reference["query"], reference["train"])
errors = {
"scalar_jacobian_vs_alias_sum_max_abs": max(
max(values.values()) for values in gradient_errors.values()
),
"kernel_vs_common_plus_diagonal_plus_cross_abs": abs(actual - common - diagonal - cross),
"untied_kernel_vs_common_plus_diagonal_abs": abs(independent_actual - common - diagonal),
"kernel_difference_vs_cross_terms_abs": abs(actual - independent_actual - cross),
}
if not all(error < TOLERANCE for error in errors.values()):
raise AssertionError(errors)
embedding = "transformer.wte.weight"
return {
"arm": arm,
"scalar_outputs": scalar_values,
"unique_parameters": sum(parameter.numel() for parameter in model.parameters()),
"parameter_aliases": aliases,
"kernel": actual,
"untied_kernel": independent_actual,
"common_parameter_kernel": common,
"common_tied_embedding_kernel": float(
(reference["query"][embedding] * reference["train"][embedding]).sum()
),
"block_diagonal_kernel": diagonal,
"block_cross_call_kernel": cross,
"per_parameter_jacobian_errors": gradient_errors,
"errors": errors,
"passed": True,
}, gradients
def check_plain_sgd(model, gradients, kernel):
"""Check Eq. 26 for 1/2 * (one train logit - 0.5)**2 loss."""
before = float(scalar_output(model, "query").detach())
derivative = float(scalar_output(model, "train").detach()) - 0.5
parameters = dict(model.named_parameters())
direction = {name: -derivative * gradient for name, gradient in gradients["train"].items()}
direct_derivative = dot(gradients["query"], direction)
derivative_error = abs(direct_derivative + derivative * kernel)
if derivative_error >= TOLERANCE:
raise AssertionError({"SGD_first_derivative_error": derivative_error})
records = []
for step in SGD_STEPS:
disposable = copy.deepcopy(model)
with torch.no_grad():
for name, parameter in disposable.named_parameters():
parameter.copy_(parameters[name] + step * direction[name])
change = float(scalar_output(disposable, "query")) - before
prediction = step * direct_derivative
residual = abs(change - prediction)
records.append(
{
"step_size": step,
"actual_query_logit_change": change,
"kernel_first_order_change": prediction,
"absolute_taylor_remainder": residual,
"absolute_remainder_divided_by_step_squared": residual / step**2,
}
)
ratios = [
records[index]["absolute_taylor_remainder"]
/ records[index + 1]["absolute_taylor_remainder"]
for index in range(len(records) - 1)
]
# A halved step has quarter-sized quadratic error. These bounds only assess
# the fixed, nondegenerate tiny numerical example; they are not a proof of
# a global remainder bound or a rule for selecting a training learning rate.
passed = all(3.0 < ratio < 5.0 for ratio in ratios)
if not passed:
raise AssertionError({"finite_difference_records": records, "ratios": ratios})
return {
"loss": "0.5 * (selected train token logit - 0.5)**2",
"optimizer": "plain SGD, no momentum, clipping, decay or optimizer state",
"kernel_directional_derivative_error": derivative_error,
"steps": records,
"remainder_ratio_when_step_halved": ratios,
"accepted_ratio_interval": [3.0, 5.0],
"passed": passed,
}
def main():
parser = argparse.ArgumentParser(description=__doc__)
parser.add_argument("--output", type=Path, default=Path(__file__).with_suffix(".json"))
args = parser.parse_args()
torch.set_num_threads(1)
torch.manual_seed(SEED)
models = {
arm: construct({**SPEC, "arm": arm}, "cpu").double().eval()
for arm in ("shared", "untied", "mlp_shared")
}
untied = models["untied"]
reference = {name: scalar_jacobian(untied, name)[1] for name in OUTPUTS}
with torch.no_grad():
tokens = torch.tensor([OUTPUTS[name]["tokens"][0] for name in OUTPUTS])
baseline = untied(tokens)
forward_errors = {arm: max_abs(model(tokens) - baseline) for arm, model in models.items()}
if not all(error < TOLERANCE for error in forward_errors.values()):
raise AssertionError(forward_errors)
checks = {}
for arm, model in models.items():
check, gradients = check_kernel(model, untied, reference, arm)
check["plain_sgd_taylor_check"] = check_plain_sgd(model, gradients, check["kernel"])
checks[arm] = check
files = [
Path(__file__),
ROOT / "src/llm_memory_editability/loop_learning.py",
ROOT / "src/llm_memory_editability/grokking_reproduction.py",
]
result = {
"created_utc": datetime.now(timezone.utc).isoformat(),
"purpose": "Independent numerical verification of theory Eq. 24-26; no science run changes",
"device": "cpu",
"dtype": "torch.float64",
"python": platform.python_version(),
"torch": torch.__version__,
"transformers": transformers.__version__,
"spec": SPEC,
"scalar_selections": OUTPUTS,
"absolute_error_tolerance": TOLERANCE,
"initial_full_logits_max_abs_errors": forward_errors,
"checks": checks,
"source_sha256": {
str(path.relative_to(ROOT)): hashlib.sha256(path.read_bytes()).hexdigest()
for path in files
},
"passed": True,
"limits": (
"Tiny randomly initialized complete ReproductionGPT with causal attention, "
"GELU MLP, residual paths, LayerNorm and tied input/output embedding. "
"Kernel identities hold at the shared common-function point. Finite-difference "
"steps use a fixed scalar squared loss and disposable CPU copies. No claim "
"about AdamW dynamics, knowledge quality, generalization, sample efficiency, "
"grokking, scientific training completion or an active GPU run is tested."
),
}
args.output.write_text(json.dumps(result, indent=2) + "\n")
print(
json.dumps(
{
"passed": True,
"output": str(args.output),
"absolute_error_tolerance": TOLERANCE,
"initial_forward_errors": forward_errors,
"kernel_errors": {arm: check["errors"] for arm, check in checks.items()},
"quadratic_remainder_ratios": {
arm: check["plain_sgd_taylor_check"]["remainder_ratio_when_step_halved"]
for arm, check in checks.items()
},
},
indent=2,
)
)
if __name__ == "__main__":
main()
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