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Differentially Private PCA

This notebook documents making a differentially private PCA release.

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from opendp.mod import enable_features
enable_features("contrib", "floating-point", "honest-but-curious")

import numpy as np

def sample_microdata(*, num_columns=None, num_rows=None, cov=None):
    cov = cov or sample_covariance(num_columns)
    microdata = np.random.multivariate_normal(
        np.zeros(cov.shape[0]), cov, size=num_rows or 100_000
    )
    microdata -= microdata.mean(axis=0)
    return microdata

def sample_covariance(num_features):
    A = np.random.uniform(0, num_features, size=(num_features, num_features))
    return A.T @ A

In this notebook we’ll be working with an example dataset generated from a random covariance matrix.

num_columns = 4
num_rows = 10_000
example_dataset = sample_microdata(num_columns=num_columns, num_rows=num_rows)

Releasing a DP PCA model with the OpenDP Library is easy because it provides an API similar to scikit-learn:

import opendp.prelude as dp

model = dp.sklearn.PCA(
    epsilon=1.,
    row_norm=1.,
    n_samples=num_rows,
    n_features=4,
)

A private release occurs when you fit the model to the data.

model.fit(example_dataset)

PCA(epsilon=1.0, n_components=4, n_features=4, n_samples=10000, row_norm=1.0)

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The fitted model can then be introspected just like Scikit-Learn’s non-private PCA:

print("singular values", model.singular_values_)
print("components")
model.components_

singular values [15.40825945 30.95765559 51.64750761 78.25285485]
components

array([[ 0.32635704,  0.63916974,  0.62412528,  0.30890252],
       [ 0.84399485,  0.11060222, -0.5202029 , -0.06948945],
       [-0.42549121,  0.70204137, -0.557553  ,  0.12340906],
       [ 0.01100033, -0.29388281, -0.17026812,  0.94048958]])

Instead of fitting the model, you could instead retrieve the measurement used to make the release, just like other OpenDP APIs. This time, we’ll also only fit 2 components. Because of this, more budget will be allocated to estimating each eigenvector internally.

model = dp.sklearn.PCA(
    epsilon=1.,
    row_norm=1.,
    n_samples=num_rows,
    n_features=4,
    n_components=2 # only estimate 2 of 4 components this time
)
meas = model.measurement()

The measurement fits model and then returns model:

meas(example_dataset)

PCA(epsilon=1.0, n_components=2, n_features=4, n_samples=10000, row_norm=1.0)

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.measurement() makes it more convenient to use the Scikit-Learn API with other combinators, like compositors.

print("singular values", model.singular_values_)
print("components")
model.components_

singular values [15.70942634 30.92864075]
components

array([[ 0.54788797,  0.64408591,  0.36746136,  0.38722636],
       [ 0.66629274, -0.06883045, -0.73004368, -0.13547169]])

Please reach out on Slack if you need to a more tailored analysis: there are lower-level APIs for estimating only the eigenvalues or eigenvectors, or to avoid mean estimation when your data is already bounded.