Isolation Forest Example

Suppose you are explaining a rejected application. The classifier has learned two different ways to become approved:

• a tiny pocket close to the query, created by a single unusual training point,

• and a larger dense approved region farther to the right.

If we ask only for the shortest valid counterfactual, the optimizer goes to the tiny pocket. Adding an isolation forest changes the feasible set: the counterfactual must still flip the prediction, but it must also stay in a region that looks typical with respect to the training data.

The figure below comes from examples/isolation_forest_example.py. It shows the MIP geometry for four thresholds. The script also solves the same query with CP and prints both backends side by side.

How to read the dashed contour

For a threshold \(\tau\), the plotted contour is not always the zero level of the isolation forest. It is the threshold-specific boundary

\[\texttt{decision\_function}(x) = -\tau - \texttt{offset\_}.\]

Points on the dense side of that curve satisfy the isolation cutoff; points on the other side are too isolated for that threshold.

This is why the four panels differ:

• 0.9 is weak, so the tiny pocket is still allowed.

• 0.51 is already strong enough to move the counterfactual inward.

• 0.5 is the historical cutoff and now excludes the tiny pocket.

• 0.1 is so strict that no target-class point survives.

Backend comparison

MIP and CP on the same query

Case	MIP status	MIP counterfactual	MIP \(L_1\)	MIP isolation score	CP status	CP counterfactual	CP \(L_1\)	CP isolation score
Plain	OPTIMAL	[0.331773, 0.864821]	0.546594	-0.050441	OPTIMAL	[0.331773, 0.864821]	0.546594	-0.050441
Isolation = 0.9	OPTIMAL	[0.331773, 0.864821]	0.546594	-0.050441	OPTIMAL	[0.331773, 0.864821]	0.546594	-0.050441
Isolation = 0.51	OPTIMAL	[0.640802, 1.169832]	1.160633	-0.009963	OPTIMAL	[0.640802, 1.169832]	1.160633	-0.009963
Isolation = 0.5	OPTIMAL	[1.021789, 1.188354]	1.560143	0.000561	OPTIMAL	[1.021789, 1.188354]	1.560143	0.000561
Isolation = 0.1	INFEASIBLE	-	-	-	INFEASIBLE	-	-	-

Why 0.1 is infeasible

This is not just a solver status. We can prove it on this example.

Let \(L_t(x)\) be the path length of the leaf reached by \(x\) in isolation tree \(t\). Let

\[L(x) = \sum_{t \in \mathcal{T}_{iso}} L_t(x)\]

be the total isolation-path length over the parsed isolation trees.

Keep the target-class constraints, remove the threshold by setting isolation_threshold=1.0, and maximize the same aggregate isolation-path length expression used by the MIP and CP backends. On this dataset, the best target-class point reaches

\[\max L(x) = 514.377459.\]

At isolation_threshold=0.1, the model requires

\[L(x) \ge 1559.821878.\]

Therefore

\[514.377459 < 1559.821878,\]

so no target-class point can satisfy the isolation inequality. The feasible set is empty for both MIP and CP because they enforce the same path-length cutoff over the same parsed isolation trees.

Run the example

python examples/isolation_forest_example.py

The script saves the figure directly to docs/_static/figures/isolation-forest-example-2d.svg and prints the full comparison table together with the infeasibility proof.