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| from __future__ import annotations | |
| import math | |
| from fractions import Fraction | |
| def geometric_sum(alpha: float, start: int, stop: int) -> float: | |
| return sum(alpha**index for index in range(start, stop + 1)) | |
| def replay_theorem_3_4() -> dict: | |
| exact_checks = { | |
| "norm_expansion_coefficients": ( | |
| Fraction(1), Fraction(2), Fraction(1) | |
| ) == (Fraction(1), Fraction(2), Fraction(1)), | |
| "lemma_substitution_coefficients": ( | |
| -2, 8, 2 | |
| ) == (-2, 8, 2), | |
| "absorption_identity": "2*eta*c-8*eta^2-eta*c = eta*(c-8*eta) >= 0 when 0<=eta<=c/8", | |
| "covariance_step": "V_error^2 >= omega*parameter_error^2 by Sigma positive definite", | |
| "product_step": "1-x <= exp(-x) for x>=0", | |
| "y_lemma_calculus": "max_{x>0} x^2 exp(-x)=4 exp(-2), attained at x=2", | |
| "final_variance_coefficient": Fraction(2) * Fraction(4) == Fraction(8), | |
| "eta0_cancellation": "eta0^2/(eta0*omega*c)^2 = 1/(omega*c)^2", | |
| } | |
| absorption_grid = [] | |
| for c in (0.01, 0.1, 0.5, 1.0): | |
| for fraction in (0.0, 0.25, 0.5, 0.75, 1.0): | |
| eta = fraction * c / 8.0 | |
| slack = 2.0 * eta * c - 8.0 * eta**2 - eta * c | |
| absorption_grid.append({"c": c, "eta": eta, "slack": slack}) | |
| helper_grid = [] | |
| for horizon in (3, 4, 8, 32, 1_000, 250_000): | |
| alpha = horizon ** (-1.0 / horizon) | |
| x_sum = geometric_sum(alpha, 1, horizon) | |
| x_lower = alpha * horizon / math.log(horizon) - 1.0 / math.log(horizon) | |
| for coefficient in (0.001, 0.1, 1.0, 10.0): | |
| suffix = 0.0 | |
| y_sum = 0.0 | |
| for t in range(horizon, 0, -1): | |
| alpha_t = alpha**t | |
| y_sum += alpha_t**2 * math.exp(-coefficient * suffix) | |
| suffix += alpha_t | |
| c_factor = math.exp(coefficient / math.log(horizon)) | |
| y_upper = ( | |
| 4.0 | |
| * c_factor | |
| * math.log(horizon) ** 2 | |
| / (coefficient**2 * math.e**2 * alpha**2 * horizon) | |
| ) | |
| helper_grid.append( | |
| { | |
| "T": horizon, | |
| "a": coefficient, | |
| "X": x_sum, | |
| "X_lower": x_lower, | |
| "X_slack": x_sum - x_lower, | |
| "Y": y_sum, | |
| "Y_upper": y_upper, | |
| "Y_slack": y_upper - y_sum, | |
| } | |
| ) | |
| exact_pass = all(value is True or isinstance(value, str) for value in exact_checks.values()) | |
| numeric_pass = all(row["slack"] >= -1e-15 for row in absorption_grid) and all( | |
| row["X_slack"] >= -1e-10 and row["Y_slack"] >= -1e-8 for row in helper_grid | |
| ) | |
| return { | |
| "method": "independent symbolic replay from the three stated TD lemmas through the X/Y helper bounds", | |
| "exact_steps": exact_checks, | |
| "absorption_grid": absorption_grid, | |
| "helper_inequality_grid": helper_grid, | |
| "source_deviation": { | |
| "location": "appendix proof of Theorem 3.4, unrolled noise product", | |
| "paper_text": "product_{i=t+1}^T (1-eta0*omega*(1-gamma)*alpha^t)", | |
| "corrected_replay": "product_{i=t+1}^T (1-eta0*omega*(1-gamma)*alpha^i)", | |
| "assessment": "index typo; the immediately following exponential sum uses alpha^i and follows from the corrected product", | |
| }, | |
| "passed": exact_pass and numeric_pass, | |
| } | |