Aggregation: Mean

Any constructors that have not completed the proof-writing and vetting process may still be accessed if you opt-in to “contrib”. Please contact us if you are interested in proof-writing. Thank you!

import opendp.prelude as dp
dp.enable_features("contrib")

Known Dataset Size

The much easier case to consider is when the dataset size is known:

input_space = dp.vector_domain(dp.atom_domain(bounds=(0., 10.)), size=10), dp.symmetric_distance()
sb_mean_trans = dp.t.make_mean(*input_space)
sb_mean_trans([5.] * 10)

5.0

The sensitivity of this transformation is the same as in make_sum (when dataset size is known), but is divided by size.

That is, \(map(d_{in}) = (d_{in} // 2) \cdot max(|L|, U) / size\), where \(//\) denotes integer division with truncation.

# since we are in the bounded-DP model, d_in should be a multiple of 2,
# because it takes one removal and one addition to change one record
sb_mean_trans.map(2)

1.0000000000000169

Note that this operation does not divide by the length of the input data, it divides by the size parameter passed to the constructor. As in any other context, it is expected that the data passed into the function is a member of the input domain, so no promises of privacy or correctness are guaranteed when the data is not in the input domain. In particular, the function may give a result with no error message.

sb_mean_trans = dp.t.make_mean(*input_space)
sb_mean_trans([5.])

0.5

You can check that a dataset is a member of a domain by calling .member:

sb_mean_trans.input_domain.member([5.])

False

In this case, [5.] is not a member because the input domain consists of vectors of length ten.

Unknown Dataset Size

There are several approaches for releasing the mean when the dataset size is unknown.

The first approach is to use the resize transformation. You can separately release an estimate for the dataset size, and then preprocess the dataset with a resize transformation.

data = [5.] * 10
bounds = (0., 10.)

input_space = dp.vector_domain(dp.atom_domain(T=float)), dp.symmetric_distance()

# (where TIA stands for Atomic Input Type)
count_meas = input_space >> dp.t.then_count() >> dp.m.then_laplace(1.)

dp_count = count_meas(data)

mean_meas = (
    input_space >>
    dp.t.then_clamp(bounds) >>
    dp.t.then_resize(dp_count, constant=5.) >>
    dp.t.then_mean() >>
    dp.m.then_laplace(1.)
)

mean_meas(data)

6.862637830873848

The total privacy expenditure is the composition of the count_meas and mean_meas releases.

dp.c.make_basic_composition([count_meas, mean_meas]).map(1)

2.000000000000017

Another approach is to compute the DP sum and DP count, and then postprocess the output.

dp_sum = input_space >> dp.t.then_clamp(bounds) >> dp.t.then_sum() >> dp.m.then_laplace(10.)
dp_count = input_space >> dp.t.then_count() >> dp.m.then_laplace(1.)

dp_fraction_meas = dp.c.make_basic_composition([dp_sum, dp_count])

dp_sum, dp_count = dp_fraction_meas(data)
print("dp mean:", dp_sum / dp_count)
print("epsilon:", dp_fraction_meas.map(1))

dp mean: 7.778118283305409
epsilon: 2.000000009313226

The same approaches are valid for the variance estimator. The Unknown Dataset Size notebook goes into greater detail on the tradeoffs of these approaches.