fix: make prv accountant robust to larger epsilons

[Solosneros]

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Docs change / refactoring / dependency upgrade

Motivation and Context / Related issue

Hi,

this PR fixes #601 and #604.

It will introduce the same fix as in microsoft/prv_accountant#38. Lukas (author of prv accountant, @wulu473) said that In general, adding any additional points is safe and won't affect the robustness negatively.

The cause of these errors seems to be the grid for computing the mean() function of the PrivacyRandomVariableTruncated class. The grid (points variable) used to compute the mean is constant apart from the lowest (self.t_min) and highest point (self.t_max).

This PR determines the grid (points variable) based on the lowest and highest point. More information is below.

Best

Observation

I debugged the code and arrived at some point at the mean() function of the PrivacyRandomVariableTruncated class. The grid (points variable) used to compute the mean is constant apart from the lowest (self.t_min) and highest point (self.t_max). See the line of code here. It looks like this [self.tmin, -0.1, -0.01, -0.001, -0.0001, -1e-05, 1e-05, 0.0001, 0.001, 0.01, 0.1, self.tmax].

It seems that the tmin and tmax are of the order of [-12,12] for the examples that I posted above and even up to [-48,48] for the example that @jeandut posted in the #604 issue whereas they are more like [-7,7] for the readme example for DP-SGD.

We suspect that the integration breaks down when the gridspacing between between tmin / tmax get's too large.

Proposed solution

Determine the points grid based on tmin and tmax but determines the start and end of the logspace based on tmin and tmax.

Before: (

opacus/opacus/accountants/analysis/prv/prvs.py

Lines 99 to 106 in 95df090

	points = np.concatenate(
	[
	[self.t_min],
	-np.logspace(-5, -1, 5)[::-1],
	np.logspace(-5, -1, 5),
	[self.t_max],
	]
	)

)

After:

# determine points based on t_min and t_max
lower_exponent = int(np.log10(np.abs(self.t_min)))
upper_exponent = int(np.log10(self.t_max))
points = np.concatenate(
    [
        [self.t_min],
        -np.logspace(start=lower_exponent, stop=-5, num=10),
        [0],
        np.logspace(start=-5, stop=upper_exponent, num=10),
        [self.t_max],
    ]
)

How Has This Been Tested (if it applies)

I ran the examples from the issues #601 and #604 and they don't break anymore.

import opacus
target_delta = 0.001
target_epsilon = 20
steps = 5000
sample_rate=0.19120458891013384

for target_epsilon in [20, 50]:
    noise_multiplier = opacus.privacy_engine.get_noise_multiplier(target_delta=target_delta, target_epsilon=target_epsilon, steps=steps, sample_rate=sample_rate, accountant="prv")
    prv_accountant = opacus.accountants.utils.create_accountant("prv")
    prv_accountant.history = [(noise_multiplier, sample_rate, steps)]
    obtained_epsilon = prv_accountant.get_epsilon(delta=target_delta)
    print(f"target epsilon {target_epsilon}, obtained epsilon {obtained_epsilon}")

target epsilon 20, obtained epsilon 19.999332284974717
target epsilon 50, obtained epsilon 49.99460075990896

target_epsilon = 4
batch_size = 50
epochs = 5
delta = 1e-05
expected_len_dataloader = 500 // batch_size
sample_rate = 1/expected_len_dataloader


noise_multiplier = opacus.privacy_engine.get_noise_multiplier(target_delta=target_delta, target_epsilon=target_epsilon, epochs=epochs, sample_rate=sample_rate, accountant="prv")
prv_accountant = opacus.accountants.utils.create_accountant("prv")
prv_accountant.history = [(noise_multiplier, sample_rate, int(epochs / sample_rate))]
obtained_epsilon = prv_accountant.get_epsilon(delta=target_delta)
print(f"target epsilon {target_epsilon}, obtained epsilon {obtained_epsilon}")

target epsilon 4, obtained epsilon 3.9968389923130356

Checklist

• The documentation is up-to-date with the changes I made.
• I have read the CONTRIBUTING document and completed the CLA (see CONTRIBUTING).
• All tests passed, and additional code has been covered with new tests.

Not able to run all tests locally and unsure if new tests should be added.

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This pull request has been merged in ad084da.