AI Explainability: Counterfactual Recourse

Applicant (input point)

Feature x₁ — income score 0.30

Feature x₂ — repayment history 0.30

Model prediction

P(approve)

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Decision

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Feature attribution (∂P/∂x · x)

x₁

0.00

x₂

0.00

Counterfactual search

Sparsity weight λ 0.30

Steps moved

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‖Δx‖ (recourse cost)

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How it works

The model is a small logistic-regression classifier over two features: P(approve) = σ(w₁x₁ + w₂x₂ + b), σ(z) = 1/(1+e^−z). The 3D surface plots this probability over the whole (x₁, x₂) plane — its shape is exactly a sigmoid ridge whose steepest slope runs perpendicular to the decision boundary w₁x₁ + w₂x₂ + b = 0.

Attribution (input × gradient):
  ∂P/∂x_i = P(1−P)·w_i
  attribution_i = x_i · ∂P/∂x_i

Counterfactual objective (Wachter et al. 2018):
  minimize  (P(x') − target)² + λ‖x' − x₀‖²
  x' ← x' − η · ∇ₓ' [loss]   (gradient descent)

Feature attribution (top box) answers "why did the model decide this?" — it is a local, first-order explanation at the current point. Counterfactual search answers a different question: "what is the smallest change to the inputs that would flip the decision?" It runs real gradient descent on the objective above, visualized as a glowing trail climbing the probability surface toward the 50% contour. A larger λ penalizes moving far from the original point, so the search prefers a smaller, more actionable change — this is the core idea behind algorithmic recourse: not just explaining a decision, but telling someone what would need to change to get a different one.

x₁ / x₂ sliders — move the applicant point directly; the surface, probability and attribution bars update live.
Find Counterfactual — runs gradient descent from the current point toward the opposite decision, animates the path as beads climbing the surface.
Apply — snaps the applicant point to the counterfactual result found.
New model — redraws random weights w₁, w₂, b, reshaping the whole probability surface.

Applicant (current point) Counterfactual search path Decision boundary (P = 0.5)

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