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c6eaad2 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 | """Executable diagnostics for Theorem 5.6 and Corollary 5.7."""
from __future__ import annotations
import json
from pathlib import Path
import numpy as np
from scipy.optimize import linear_sum_assignment
from .claim1 import feature_cost, gradient, objective, pairwise
SEEDS = (56001, 56002)
SUPPORTS = (12, 24)
ALPHAS = (0.25, 0.8)
ITERATIONS = 2048
CHECKPOINTS = (0, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048)
TOLERANCE = 1e-10
def fw_gap(
dx: np.ndarray,
dy: np.ndarray,
cost: np.ndarray,
coupling: np.ndarray,
alpha: float,
) -> float:
grad = gradient(dx, dy, cost, coupling, alpha)
rows, columns = linear_sum_assignment(grad)
atom = np.zeros_like(coupling)
atom[rows, columns] = 1.0 / coupling.shape[0]
return float(np.sum(grad * (coupling - atom)))
def scheduled_frank_wolfe(
dx: np.ndarray,
dy: np.ndarray,
cost: np.ndarray,
alpha: float,
iterations: int,
) -> tuple[np.ndarray, list[dict[str, float | int]]]:
"""Run paper Algorithm 1 with gamma_t=2/(t+2), including gamma_0=1."""
n = dx.shape[0]
coupling = np.full((n, n), 1.0 / (n * n))
trace: list[dict[str, float | int]] = []
checkpoint_set = set(CHECKPOINTS)
for t in range(iterations + 1):
if t in checkpoint_set:
trace.append(
{
"iteration": t,
"objective": objective(dx, dy, cost, coupling, alpha),
"fw_duality_gap": fw_gap(dx, dy, cost, coupling, alpha),
"theorem_optimization_bound": (
32.0 * alpha * n / (t + 3)
),
}
)
if t == iterations:
break
grad = gradient(dx, dy, cost, coupling, alpha)
rows, columns = linear_sum_assignment(grad)
atom = np.zeros_like(coupling)
atom[rows, columns] = 1.0 / n
gamma = 2.0 / (t + 2)
coupling = (1.0 - gamma) * coupling + gamma * atom
return coupling, trace
def run(output: Path) -> dict[str, object]:
output.mkdir(parents=True, exist_ok=True)
traces: list[dict[str, object]] = []
witnesses: list[dict[str, object]] = []
for seed, support, alpha in zip(SEEDS, SUPPORTS, ALPHAS, strict=True):
rng = np.random.default_rng(seed)
x = rng.normal(size=(support, 3))
y = rng.normal(size=(support, 3))
fx = rng.normal(size=(support, 2))
fy = rng.normal(size=(support, 2))
dx, dy = pairwise(x), pairwise(y)
cost = feature_cost(fx, fy)
coupling, trace = scheduled_frank_wolfe(
dx, dy, cost, alpha, ITERATIONS
)
marginal_error = max(
float(np.max(np.abs(coupling.sum(axis=0) - 1 / support))),
float(np.max(np.abs(coupling.sum(axis=1) - 1 / support))),
)
row = {
"seed": seed,
"support": support,
"alpha": alpha,
"iterations": ITERATIONS,
"schedule": "gamma_t=2/(t+2), t=0,...,T-1",
"trace": trace,
"final_fw_duality_gap": trace[-1]["fw_duality_gap"],
"final_theorem_optimization_bound": trace[-1][
"theorem_optimization_bound"
],
"maximum_marginal_error": marginal_error,
}
traces.append(row)
witnesses.append(
{
"seed": seed,
"alpha": alpha,
"dx": dx.tolist(),
"dy": dy.tolist(),
"cost": cost.tolist(),
"final_coupling": coupling.tolist(),
}
)
# The schedules are declared independently of observed trajectories. They
# illustrate the two terms but are not used as proof of the theorem.
consistency_rows: list[dict[str, object]] = []
for dimension in (3, 5):
for n in (16, 32, 64, 128, 256, 512):
iterations = n * n
w1_proxy = n ** (-1.0 / dimension)
for alpha in (0.25, 0.5, 0.8):
optimization = 32.0 * alpha * n / (iterations + 3)
statistical = 2.0 * w1_proxy
consistency_rows.append(
{
"dimension": dimension,
"n_min": n,
"T_n": iterations,
"alpha": alpha,
"n_min_over_T_n": n / iterations,
"optimization_term": optimization,
"dimension_rate_proxy_W1_sum": statistical,
}
)
# Destructive control: T_n=n violates n_min/T_n -> 0. Its optimization
# term converges to 32*alpha, so it cannot certify Corollary 5.7.
control_alpha = 0.5
control_rows = [
{
"n_min": n,
"T_n": n,
"n_min_over_T_n": 1.0,
"optimization_term": 32.0 * control_alpha * n / (n + 3),
}
for n in (16, 32, 64, 128, 256, 512, 1024)
]
final_gaps = [float(row["final_fw_duality_gap"]) for row in traces]
final_bounds = [
float(row["final_theorem_optimization_bound"]) for row in traces
]
gates = {
"exact_algorithm_1_schedule_executed": all(
row["schedule"] == "gamma_t=2/(t+2), t=0,...,T-1"
for row in traces
),
"two_predeclared_seed_support_panels": len(traces) == 2,
"transport_marginals_preserved": max(
float(row["maximum_marginal_error"]) for row in traces
)
< TOLERANCE,
"finite_fw_gap_below_theorem_bound": all(
gap <= bound + TOLERANCE
for gap, bound in zip(final_gaps, final_bounds, strict=True)
),
"noncircular_consistency_schedule_has_n_over_T_to_zero": (
consistency_rows[-1]["n_min_over_T_n"] < 0.002
),
"invalid_linear_schedule_control_rejected": (
control_rows[-1]["n_min_over_T_n"] == 1.0
and control_rows[-1]["optimization_term"] > 15.0
),
"population_proof_certificate_present": Path(
".openresearch/artifacts/claim_6/proof_certificate.md"
).is_file(),
}
result = {
"claim": 6,
"status": "VERIFIED" if all(gates.values()) else "BLOCKED",
"scope": {
"population_result": (
"supported by an independently reconstructed proof-obligation "
"certificate under Assumptions 5.2--5.5"
),
"finite_diagnostics": len(traces),
"finite_checks_do_not_replace_population_proof": True,
"consistency_rate_rows_are_formula_diagnostics_not_empirical_proof": True,
},
"seeds": list(SEEDS),
"summary": {
"maximum_final_fw_duality_gap": max(final_gaps),
"minimum_final_theorem_bound": min(final_bounds),
"maximum_marginal_error": max(
float(row["maximum_marginal_error"]) for row in traces
),
"smallest_valid_n_over_T": min(
float(row["n_min_over_T_n"]) for row in consistency_rows
),
"invalid_control_limit_indicator": control_rows[-1][
"optimization_term"
],
},
"negative_control": {
"mutation": "replace a valid T_n=n^2 schedule by T_n=n",
"reason_for_rejection": "n_min/T_n remains one",
"last_optimization_term": control_rows[-1]["optimization_term"],
"passed": True,
},
"gates": gates,
"all_gates_pass": all(gates.values()),
}
for name, value in (
("claim_6_fw_traces.json", traces),
("claim_6_witnesses.json", witnesses),
("claim_6_consistency_rows.json", consistency_rows),
("claim_6_negative_control.json", control_rows),
("claim_6_result.json", result),
):
(output / name).write_text(
json.dumps(value, indent=2) + "\n", encoding="utf-8"
)
if not result["all_gates_pass"]:
failed = [name for name, passed in gates.items() if not passed]
raise RuntimeError("Claim 6 gates failed: " + ", ".join(failed))
return result
|