Transformations#
(See also opendp.transformations in the API reference.)
This section gives a high-level overview of the transformations that are available in the library. Refer to the Transformation section for an explanation of what a transformation is.
As covered in the Chaining section, the intermediate domains need to match when chaining. Each transformation has a carefully chosen input domain and output domain that supports their relation.
Note
If you pass information collected directly from the dataset into constructors, the differential privacy guarantees may be compromised. Constructor arguments should be either:
Public information, like information from a codebook or prior domain expertise
Other DP releases on the data
Preprocessing is the series of transformations that shape the data into a domain that is conformable with the aggregator.
You will need to choose the proper transformations from the sections below in order to chain with the aggregator you intend to use. The sections below are in the order you would typically chain transformations together, but you may want to peek at the aggregator section at the end first, to identify the input domain that you’ll need to preprocess to.
Casting#
Any time you want to convert between data types, you’ll want to use a casting transformation. In particular, in pipelines that load dataframes from CSV files, it is very common to cast from Strings to some other type.
Depending on the caster you choose, the output data may be null and you will be required to chain with an imputer.
Caster |
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Output Domain |
Input/Output Metric |
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Imputation#
Null values are tricky to handle in a differentially private manner. If we were to allow aggregations to propagate null, then aggregations provide a non-differentially-private bit revealing the existence of nullity in the dataset. If we were to implicitly drop nulls from sized aggregations (like a dataset mean where we divide by the number of non-null elements), then the sensitivity of non-null individuals is underestimated. Therefore, most aggregators must be fed completely non-null data. We can ensure data is non-null by imputing.
When you cast with opendp.transformations.make_cast() or opendp.transformations.make_cast_default(),
the cast may fail, so the output domain may include null values (OptionDomain or AtomDomain with nullity).
We have provided imputation transformations to transform the data domain to the non-null VectorDomain<AtomDomain<TA>>.
You may also be in a situation where you want to bypass dataframe loading and casting because you already have a vector of floats loaded into memory. In this case, you should start your chain with an imputer if the floats are potentially null.
- OptionDomain:
A representation of nulls using an Option type (
Option<bool>,Option<i32>, etc).- AtomDomain:
A representation of nulls using the data type itself (
f32andf64).
The opendp.transformations.make_impute_constant() transformation supports imputing on either of these representations of nullity,
so long as you pass the DA (atomic domain) type argument.
Imputer |
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Output Domain |
Input/Output Metric |
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Indexing#
Indexing operations provide a way to relabel categorical data, or bin numeric data into categorical data.
These operations work with usize data types: an integer data type representing an index.
opendp.transformations.make_find() finds the index of each input datum in a set of categories.
In other words, it transforms a categorical data vector to a vector of numeric indices.
>>> import opendp.prelude as dp
>>> dp.enable_features('contrib')
>>> finder = (
... # define the input space
... (dp.vector_domain(dp.atom_domain(T=str)), dp.symmetric_distance()) >>
... # find the index of each input datum in the categories list
... dp.t.then_find(categories=["A", "B", "C"]) >>
... # impute any input datum that are not a part of the categories list as 3
... dp.t.then_impute_constant(3)
... )
>>> finder(["A", "B", "C", "A", "D"])
[0, 1, 2, 0, 3]
opendp.transformations.make_find_bin() is a binning operation that transforms numerical input data to a vector of bin indices.
>>> binner = dp.t.make_find_bin(
... dp.vector_domain(dp.atom_domain(T=float)), dp.symmetric_distance(),
... edges=[1., 2., 10.])
>>> binner([0., 1., 3., 15.])
[0, 1, 2, 3]
opendp.transformations.make_index() uses each indicial input datum as an index into a category set.
>>> indexer = dp.t.make_index(
... dp.vector_domain(dp.atom_domain(T=dp.usize)), dp.symmetric_distance(),
... categories=["A", "B", "C"], null="D")
>>> indexer([0, 1, 2, 3, 2342])
['A', 'B', 'C', 'D', 'D']
You can use combinations of the indicial transformers to map hashable data to integers, bin numeric types, relabel hashable types, and label bins.
Indexer |
Input Domain |
Output Domain |
Input/Output Metric |
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Clamping#
Many aggregators depend on bounded data to limit the influence that perturbing an individual may have on a query.
For example, the stability map for the opendp.transformations.make_sum() aggregator is d_out = d_in * max(|L|, U).
This relation states that adding or removing d_in records may influence the sum by d_in * the greatest magnitude of a record.
Any aggregator that needs bounded data will indicate it in the function name.
In these kinds of aggregators the relations make use of the clamping bounds L and U to translate d_in to d_out.
Clamping generally happens after casting and imputation but before resizing. Only chain with a clamp transformation if the aggregator you intend to use needs bounded data.
Clamper |
Input Domain |
Output Domain |
Input/Output Metric |
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Dataset Ordering#
Most dataset-to-dataset transformations are not sensitive to the order of elements within the dataset. This includes all row-by-row transformations. These transformations are written to operate with symmetric distances.
Transformations that are sensitive to the order of elements in the dataset use the InsertDeleteDistance metric instead. It is common for aggregators to be sensitive to the dataset ordering.
The following transformations are used to relate dataset metrics that are not sensitive to ordering (SymmetricDistance and ChangeOneDistance)
to metrics that are sensitive to ordering (InsertDeleteDistance and HammingDistance respectively).
Caster |
Input/Output Domain |
Input Metric |
Output Metric |
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Bounded Metrics#
You may be more familiar with “bounded” differential privacy, where dataset distances are expressed in terms of the number of changed rows.
Expressing dataset distances in this manner is more restrictive, as edit distances are only valid for datasets with a fixed size.
Generally speaking, if a dataset differs from a neighboring dataset by no more than k edits, then they differ by no more than 2k additions and removals.
We therefore write all transformations in terms of the more general “unbounded”-dp metrics SymmetricDistance and InsertDeleteDistance,
and provide the following constructors to convert to/from “bounded”-dp metrics ChangeOneDistance and HammingDistance respectively.
Caster |
Input/Output Domain |
Input Metric |
Output Metric |
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Resizing#
The resize transformation takes a dataset with unknown size, and a target size (that could itself be estimated with a DP count). If the dataset has fewer records than the target size, additional rows will be imputed. If the dataset has more records than the target size, a simple sample of the rows is taken.
In the case that a neighboring dataset adds one record to the dataset, and it causes one fewer imputation,
the resulting dataset distance is 2.
Therefore, the resize transformation is 2-stable: map(d_in) = 2 * d_in.
Similarly to data bounds, many aggregators calibrate their stability map based on knowledge of a known dataset size.
For example, the relation downstream for the opendp.transformations.make_mean() aggregator is map(d_in) = d_in // 2 * (U - L) / n.
Notice that any addition and removal may, in the worst case, change a record from L to U.
Such a substitution would influence the mean by (U - L) / n.
Any aggregator that needs sized data will indicate it in the function name.
In these kinds of aggregators, the relations need knowledge about the dataset size n to translate d_in to d_out.
Resizing generally happens after clamping. Only chain with a resize transformation if the aggregator you intend to use needs sized data.
The input and output metrics may be configured to any combination of SymmetricDistance and InsertDeleteDistance.
Resizer |
Input Domain |
Output Domain |
Input/Output Metric |
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Aggregators#
Aggregators compute a summary statistic on individual-level data.
Aggregators that produce scalar-valued statistics have an output_metric of AbsoluteDistance[TO].
This output metric can be chained with most noise-addition measurements interchangeably.
However, aggregators that produce vector-valued statistics like opendp.transformations.make_count_by_categories()
provide the option to choose the output metric: L1Distance[TOA] or L2Distance[TOA].
These default to L1Distance[TOA], which chains with L1 noise mechanisms like opendp.measurements.make_laplace() and opendp.measurements.make_laplace().
If you set the output metric to L2Distance[TOA], you can chain with L2 mechanisms like opendp.measurements.make_gaussian().
The constructor opendp.transformations.make_count_by() does a similar aggregation as opendp.transformations.make_count_by_categories(),
but does not need a category set (you instead chain with opendp.measurements.make_laplace_threshold()).
The make_sized_bounded_covariance aggregator is Rust-only at this time because data loaders for data of type Vec<(T, T)> are not implemented.
See the notebooks for code examples and deeper explanations:
Aggregator |
Input Domain |
Output Domain |
Input Metric |
Output Metric |
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make_sized_bounded_covariance (Rust only) |
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opendp.transformations.make_sum() makes a best guess as to which summation strategy to use based on the input space.
Should you need it, the following constructors give greater control over the sum.
Algorithm Details
The following strategies are ordered by computational efficiency:
checkedcan be used when the dataset size multiplied by the bounds doesn’t overflow.monotoniccan be used when the bounds share the same sign.orderedcan be used when the input metric isInsertDeleteDistance.splitseparately sums positive and negative numbers, and then adds these sums together.
All four algorithms are valid for integers, but only checked and ordered are available for floats.
There are separate constructors for integers and floats, because floats additionally need a dataset truncation and a slightly larger sensitivity.
The increase in float sensitivity accounts for inexact floating-point arithmetic, and is calibrated according to the length of the mantissa and underlying summation algorithm.
Floating-point summation may be further configured to either Sequential<T> or Pairwise<T> (default).
Sequential summation results in an O(n^2 / 2^k) increase in sensitivity, while pairwise summation results only in a O(log_2(n)n / 2^k)) increase,
where k is the bit length of the mantissa in the floating-point numbers used.
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Input Metric |
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Quantiles via Trees#
Building off of the binning transformation, quantiles can be estimated by privatizing a b-ary tree and then postprocessing. See the following notebook for more information:
These use opendp.transformations.make_b_ary_tree(), opendp.transformations.make_consistent_b_ary_tree() and opendp.transformations.make_quantiles_from_counts().