Poisson subsampling on Gaussian DP

[kwunhangwong]

Description

Hello there, I have been trying to reimplement Gaussian Differential Privacy by myself.

In your code of Gaussian accountant, I saw the subsampling amplification for computing mu, which is default adopted in your gdp accountant.
However, I could not find this equation from the original paper.

def compute_mu_poisson(
    *, steps: int, noise_multiplier: float, sample_rate: float
) -> float:
    """
    Compute mu from uniform subsampling.
    """
    return np.sqrt(np.exp(noise_multiplier ** (-2)) - 1) * np.sqrt(steps) * sample_rate

I only saw another one in original paper shown in the below, may you kindly cite the reference of the above equation?

return (
      np.sqrt(2)
              * c
              * np.sqrt(
                  np.exp(noise_multiplier ** (-2)) * norm.cdf(1.5 / noise_multiplier)
                  + 3 * norm.cdf(-0.5 / noise_multiplier)
                  - 2
              )
          )

Many thanks !!

[iden-kalemaj]

Thank you for bringing this up. The mu computation is from here: https://github.com/woodyx218/Deep-Learning-with-GDP-Pytorch/blob/master/gdp_accountant.py#L33.

However, I am also not seeing this expression in the paper, so let me take some time to see if there is a simple way to derive it.