# Counterfactuals with Constraint Solvers

## Scoring

An implementation on how to calculate counterfactuals with Constraint Solvers (namely OptaPlanner) is available here.

This implementation satisfies several criteria of the counterfactuals.

The penalisation score is represented with a BendableBigDecimalScore 1, having three “hard” levels and two “soft” levels.

The first hard level component, 1, penalises the score according to the distance between the prediction, $y^{\prime}$ for the currently proposed solution, $x^{\prime}$ and the original prediction $y$, that is this our $(\hat{f}(x^{\prime})-y^{\prime})^2$. This corresponds to the counterfactual’s validity.

The actionability is score with 2. This component penalises the score according to number of immutable features which were changed in the counterfactual.

A confidence score component, 3 is use to, optionally, impose a minimum confidence threshold to the counterfactual’s associated prediction, $x^{\prime}$.

Finally, the feature distance, 4, penalises the score according to the feature distance. This is the representation of

$$d(x, x^{\prime}).$$

In the concrete implementation linked above, the distance, $d$, chosen is a Manhattan (or $L^1$) distance calculated feature-wise. This also corresponds to the validity property.

## Implementation

Entities are defined by classes such as Integer, Categorical, Boolean or Float, as shown in 5. Each of the features, shown in 6, is created as an instance of one of these entities. For instance, feature1 would be of type Integer and feature2 would be of type Categorical, etc.

The original data point, $x$ is represented by this set of features (6).

A planning solution (PlanningSolution), illustrated in 7 will produce candidate solutions (shown in 8)

For each solution, we propose a new set of features ($x^{\prime}$) as a counterfactual candidate. For instance, ~solution A~ in 8.

In the following section we will look at how each component is calculated. We will refer to each “hard” level component as $H_1, H_2$ and $H_3$ and the “soft” component as $S_1$. The overal score consists, then, of $S={H_1, H_2, H_3, S_1 }$

### Prediction distance

The first component of the score, 1 is established by sending the proposed counterfactual $x^{\prime}$, 8 to a predictive model, 9 and calculating the distance between the desired outcome, $y^{\prime}$ and the model’s prediction. This is done component wise, for each feature of the output. That is, for a prediction with $N$ features, we calculate

$$H_1=\left(\sum_i^Nf(x^{\prime}_i) - y^{\prime}_i\right)^2$$

### Tolerance

For numerical features, the above score ($H_1$) will cause the counterfactual to be invalid, unless the distance between the outcomes and proposed values is exactly zero.

We can solve this problem by introducing a “tolerance” adjustment, which allows proposed values to be accepted if they are “close enough” to the goal.

To make the tolerance scale-invariant and unit-less we can use a relative change and set the distance to zero, if smaller than the threshold $t$, that is

$$d = \begin{cases} 0,&\qquad\text{if},\frac{\vert f(x’_i)-y’_i\vert}{\max(\vert f(x’_i)\vert,\vert y’_i\vert)} < t \\ \vert f(x’_i)-y’_i\vert,&\qquad\text{otherwise} \end{cases}$$

and compare to the threshold $t$. As an example, for a goal $y’_i=3$ and a threshold of $t=0.01$, around the goal we would have the distance as in the figure below:

This would however fail for the edge case where $y^{\prime}_i=0$ as we can see below:

To solve this, we can introduce a special case for $y^{\prime}_i=0$, such that:

$$d_{g=0} = \begin{cases} 0,&\qquad \text{if}\ |f(x^{\prime}_i)|< t\\ \lVert f(x^{\prime}_i) - y^{\prime}_i \rVert,&\qquad\text{otherwise} \end{cases}$$

So that we have now the desired behaviour at $y^{\prime}_i=0$:

### Gower distance

An alternative metric for the outcome distance (and mixed variables in general) is the site/Machine learning/Gower distance.

## Actionability score

For the second component, the actionability score, 2. We calculate the number of features for the protected set $\mathcal{A}$, which have a different value from the original. That is, assuming we have a certain number of protectd features $M$, such that $\mathcal{A}={A_1,A_2,\dots,A_M}$, we calculate:

$$H_2 = \sum_{a \in \mathcal{A}} \mathbb{1}(x_a \neq x^{\prime}_a),$$

## Confidence score

For each feature $i$, if we have a prediction confidence, $p_i(f(x^{\prime}))$, we calculate the number of features which have a confidence below a certain predefined threshold, $P_i$. If the threshold is not defined, this component will always be zero and not influence the counterfactual selection. Assuming we have defined a threshold for all $N$ features, $P = {P_1, P_2, \dots, P_N}$ we calculate this score as

$$H_3 = \sum_i^N \mathbb{1} \left( p_i \left( f(x^{\prime}) < P_i \right) \right)$$

## Feature distance

Considering that each datapoint $x$ consists of different $N$ features, such that $x=\left(f_1,\dots,f_n\right)$ and that each feature might be numerical or categorical2, we calculate the distance between a datapoint $x$ and a potential counterfactual $x^{\prime}$:

$$d\left(x,x^{\prime}\right)=\sum_{i=1}^Nd^{\prime}\left(x_i,x_i^{\prime}\right)$$ $$d^{\prime}\left(x_i,x_i^{\prime}\right)= \begin{cases} \left(x_i-x_i^{\prime}\right)^2,\quad\text{if}\ x_i,x_i^{\prime}\in\mathbb{N} \lor x_i,x_i^{\prime}\in\mathbb{R}\ 1-\delta_{x,x^{\prime}},\quad\text{if}\ x_i,x_i^{\prime}\ \text{categorical} \end{cases}$$

Since in many scenarios we might not have access to the training data, the above distance are not normalised. In the event that we do have access to training data, then we can use the standard deviation ($SD$) to normalise the features. The $SD$ can be calculated as:

$$SD=\sqrt{\frac{1}{N}\sum_{i=1}^N\left(x_i-\bar{x}\right)^2}$$

so that, in this case, we scale the numerical features with

$$\bar{d}^{\prime}\left(x_i,x_i^{\prime}\right)= \frac{\left(x_i-x_i^{\prime}\right)^2}{SD}.$$

## Searching

To search for a counterfactual, we start by specifying a search domain for each feature. This will include:

• An upper and lower bounds for numerical features, respectively $\mathcal{D}_l, \mathcal{D}_u$
• A set of categories for categorical features, $\mathcal{C}$
• $\mathcal{B}={0,1}$ for the specific case of boolean/binary values

Typically these values would be either established by someone with domain knowledge, or by values that might reflect our expectation for the actual counterfactual (for instance, an ~age~ would have realistic values).

The algorithm used for the search is Tabu search3 (Glover, 1989).

1. Here we are considering binary and boolean values as categorical. ↩︎

2. cite:glover1989tabu Glover, F. (1989). Tabu search—part i. ORSA Journal on computing, 1(3), 190–206. ↩︎