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Aggregate Functions

Aggregate functions combine the utility scores computed per attribute into a single global score used by ARX to compare and choose between different transformations.

Role

Some quality models (loss, height, precision, nm_entropy) produce a score per quasi-identifier. The aggregate function determines how these individual scores are merged into a scalar value that the optimizer can minimize.

flowchart LR
    A["score(age)"]     --> F[Aggregate function]
    B["score(sex)"]     --> F
    C["score(race)"]    --> F
    D["score(marital-status)"] --> F
    E["score(native-country)"] --> F
    F --> G[Global score\nminimized by ARX]

Available aggregate functions

arithmetic_mean — Arithmetic Mean (default)

Computes the simple average of per-attribute scores.

\[\text{global score} = \frac{1}{n} \sum_{i=1}^{n} s_i\]
  • Treats all attributes equally.
  • Sensitive to outlier scores.

geometric_mean — Geometric Mean

Computes the geometric mean of per-attribute scores.

\[\text{global score} = \left(\prod_{i=1}^{n} s_i\right)^{1/n}\]
  • Less sensitive to extreme values than the arithmetic mean.
  • Tends to penalize heavily degraded attributes more strongly.

sum — Sum

Computes the raw sum of per-attribute scores.

\[\text{global score} = \sum_{i=1}^{n} s_i\]
  • Equivalent to the arithmetic mean for ranking purposes (same preference order).
  • Amplifies the impact of the number of attributes.

maximum — Maximum

Retains only the highest score among all attributes.

\[\text{global score} = \max(s_1, s_2, \ldots, s_n)\]
  • Minimax strategy: minimizes the worst-case loss on any single attribute.
  • Favors transformations that do not excessively degrade any one attribute.

rank — Rank

Represents scores as a sorted vector and compares transformations lexicographically.

\[\text{global score} = [s_{\sigma(1)}, s_{\sigma(2)}, \ldots, s_{\sigma(n)}]\]

where \(\sigma\) sorts in descending order.

  • Compares the most degraded attributes first before breaking ties on others.
  • Multi-criteria approach without scalar aggregation.

Compatibility

Aggregate functions are only compatible with quality models that produce per-attribute scores:

Quality model Aggregate supported
loss ✓
height ✓
precision ✓
nm_entropy ✓
entropy ✗
discernibility ✗
aecs ✗
ambiguity ✗
classification ✗

Attribute weights

Individual attributes can be weighted via the attribute_weights parameter. A higher weight means ARX will accept more generalization on that attribute; a lower weight pushes towards preserving original values.

{
  "attribute_weights": {
    "age": 0.1,
    "race": 2.0
  }
}

Configuration

{
  "utility_measure": "loss",
  "utility_aggregate": "arithmetic_mean"
}

If utility_aggregate is absent or null, ARX uses its internal default.