tensorflow_privacy/privacy/analysis/rdp_accountant.py [485:536]:
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  max_alpha = 256
  assert isinstance(alpha, int)

  if np.isinf(alpha):
    return np.inf
  elif alpha == 1:
    return 0

  def cgf(x):
    # Return rdp(x+1)*x, the rdp of Gaussian mechanism is alpha/(2*sigma**2)
    return x * 1.0 * (x + 1) / (2.0 * sigma**2)

  def func(x):
    # Return the rdp of Gaussian mechanism
    return 1.0 * x / (2.0 * sigma**2)

  # Initialize with 1 in the log space.
  log_a = 0
  # Calculates the log term when alpha = 2
  log_f2m1 = func(2.0) + np.log(1 - np.exp(-func(2.0)))
  if alpha <= max_alpha:
    # We need forward differences of exp(cgf)
    # The following line is the numerically stable way of implementing it.
    # The output is in polar form with logarithmic magnitude
    deltas, _ = _get_forward_diffs(cgf, alpha)
    # Compute the bound exactly requires book keeping of O(alpha**2)

    for i in range(2, alpha + 1):
      if i == 2:
        s = 2 * np.log(q) + _log_comb(alpha, 2) + np.minimum(
            np.log(4) + log_f2m1,
            func(2.0) + np.log(2))
      elif i > 2:
        delta_lo = deltas[int(2 * np.floor(i / 2.0)) - 1]
        delta_hi = deltas[int(2 * np.ceil(i / 2.0)) - 1]
        s = np.log(4) + 0.5 * (delta_lo + delta_hi)
        s = np.minimum(s, np.log(2) + cgf(i - 1))
        s += i * np.log(q) + _log_comb(alpha, i)
      log_a = _log_add(log_a, s)
    return float(log_a)
  else:
    # Compute the bound with stirling approximation. Everything is O(x) now.
    for i in range(2, alpha + 1):
      if i == 2:
        s = 2 * np.log(q) + _log_comb(alpha, 2) + np.minimum(
            np.log(4) + log_f2m1,
            func(2.0) + np.log(2))
      else:
        s = np.log(2) + cgf(i - 1) + i * np.log(q) + _log_comb(alpha, i)
      log_a = _log_add(log_a, s)

    return log_a
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -



tensorflow_privacy/privacy/analysis/rdp_privacy_accountant.py [432:483]:
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  max_alpha = 256
  assert isinstance(alpha, int)

  if np.isinf(alpha):
    return np.inf
  elif alpha == 1:
    return 0

  def cgf(x):
    # Return rdp(x+1)*x, the rdp of Gaussian mechanism is alpha/(2*sigma**2)
    return x * 1.0 * (x + 1) / (2.0 * sigma**2)

  def func(x):
    # Return the rdp of Gaussian mechanism
    return 1.0 * x / (2.0 * sigma**2)

  # Initialize with 1 in the log space.
  log_a = 0
  # Calculates the log term when alpha = 2
  log_f2m1 = func(2.0) + np.log(1 - np.exp(-func(2.0)))
  if alpha <= max_alpha:
    # We need forward differences of exp(cgf)
    # The following line is the numerically stable way of implementing it.
    # The output is in polar form with logarithmic magnitude
    deltas, _ = _get_forward_diffs(cgf, alpha)
    # Compute the bound exactly requires book keeping of O(alpha**2)

    for i in range(2, alpha + 1):
      if i == 2:
        s = 2 * np.log(q) + _log_comb(alpha, 2) + np.minimum(
            np.log(4) + log_f2m1,
            func(2.0) + np.log(2))
      elif i > 2:
        delta_lo = deltas[int(2 * np.floor(i / 2.0)) - 1]
        delta_hi = deltas[int(2 * np.ceil(i / 2.0)) - 1]
        s = np.log(4) + 0.5 * (delta_lo + delta_hi)
        s = np.minimum(s, np.log(2) + cgf(i - 1))
        s += i * np.log(q) + _log_comb(alpha, i)
      log_a = _log_add(log_a, s)
    return float(log_a)
  else:
    # Compute the bound with stirling approximation. Everything is O(x) now.
    for i in range(2, alpha + 1):
      if i == 2:
        s = 2 * np.log(q) + _log_comb(alpha, 2) + np.minimum(
            np.log(4) + log_f2m1,
            func(2.0) + np.log(2))
      else:
        s = np.log(2) + cgf(i - 1) + i * np.log(q) + _log_comb(alpha, i)
      log_a = _log_add(log_a, s)

    return log_a
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -