Global Surrogate Models

Intuition

Train a simple, readable model (linear or a small tree) to imitate the black box. If the imitation is faithful, its structure explains the black box; if not, the explanation is fiction.

Precise explanation

1. Query the black box: $\hat y = f(X)$ on representative data.
2. Fit the surrogate $g$ to $(X, \hat y)$ — note: the black-box predictions, not the true labels.
3. Fidelity: $R^2(\hat y, g(X))$ — how much of the black box's behaviour the surrogate captures. Only interpret a surrogate with high fidelity (and check fidelity on held-out data). Fidelity says nothing about whether the black box is correct — only whether the surrogate copies it.

Linear surrogate: least squares on $[1, X]$ (NP-12).

Worked example

Black box $f(x) = 2x_0 - x_1 + 3$: linear surrogate recovers intercept 3, weights [2, −1], fidelity 1. Black box $f(x) = x_0^2$ on symmetric data: best line is flat, fidelity ≈ 0 → do not trust it.

Common mistakes

• Fitting the surrogate to true labels (that just trains another model).
• Reporting the surrogate's accuracy on the task as "fidelity".
• Interpreting a low-fidelity surrogate.

Why this matters

Surrogates are a practical way to summarise complex models, and the fidelity check is a habit every explanation method deserves.

References

• Interpretable Machine Learning (3rd edition) — Christoph Molnar (S024)