Error of randomized Milstein scheme for scalar SDEs with noisy information about coefficients and Wiener process

What the math gives to ML

The paper gives a sharp robustness law for randomized Milstein integration when both coefficient evaluations and Wiener-path increments are noisy: the strong error is the sum of a discretization term and an unavoidable linear oracle-noise floor. This transfers naturally to score-based diffusion samplers, where a neural score or drift network is only an approximate coefficient oracle and numerical Brownian increments may also be quantized or otherwise perturbed. The most useful engineering consequence is a principled rule for balancing solver steps against score and derivative accuracy: increasing the number of steps beyond the oracle-noise floor cannot improve sampling fidelity.