Question bank — Counterfactual explanations

False. Importance tells you which features mattered on average; a counterfactual tells you the specific minimal edit to $\mathbf{x}$ that flips this one prediction. Contrastive, not aggregate.

False. "Closest" only optimises proximity. If the closest edit changes an immutable feature (age, birthplace), it is mathematically valid but useless — usefulness also needs actionability and plausibility.

False. There are usually many valid ${\mathbf{x}}^{\prime }$ on the far side of the boundary; that is precisely why DiCE searches for a diverse set rather than a single point.

False. It must lie past the boundary (strictly in the target class region). The boundary point is the cheapest candidate for a linear model, but validity requires $f({\mathbf{x}}^{\prime })=y^{\prime }$, which holds strictly beyond it.

False. $L_1$ tends to concentrate change into few features (sparse edits); $L_2$ spreads a little change across many. The resulting explanations look qualitatively different — a story about one feature vs. a diffuse nudge.

False. Both minimise input change subject to flipping $f$. The math is nearly identical; the difference is intent and constraints — see Adversarial Examples. Counterfactuals add plausibility/actionability so the flip is meaningful, not a hidden pixel attack.

False. Flipping is a property of the model; acting is a property of the world. Recourse is the extra requirement that the suggested change be feasible for this person to perform.

False. Without it, gradient descent happily finds off-manifold points (negative income, impossible pixel noise) — exactly the adversarial regime. Plausibility keeps ${\mathbf{x}}^{\prime }$ inside realistic data.

The error is ignoring immutability. Age cannot be edited, so this "explanation" gives no recourse. Fix: set the change cost $w_{\text{age}}=\infty$ so the optimiser never touches it.

Smallest total change under $L_2$ typically nudges every feature a little, producing an unreadable explanation. Tabular counterfactuals should prefer $L_1$ for sparse, human-digestible edits.

Reversed. It is zero once $p>0.5$ and positive before — it stops pushing exactly when the target class is reached, which is what we want.

Actionable in principle, but the causal graph matters: obtaining a PhD also changes years-of-experience, age, and income. A Causal Inference-aware method (FACE) propagates these downstream effects instead of editing one node in isolation.

The closed form ${\mathbf{x}}^{\prime }=\mathbf{x}-\frac{{\mathbf{w}}^{\top }\mathbf{x}+b}{\Vert \mathbf{w}{\Vert }^2}\mathbf{w}$ assumes a single flat boundary. Neural boundaries curve, so we must walk toward them with gradient descent — no closed form.

Continuous optima need rounding to actionable granularity ("about $52k"). For categorical one-hot features, the analogous error is reporting fractional categories instead of rounding to a valid level.

The minus is intentional. We minimise the loss, so a minus rewards large pairwise distances — pushing the counterfactuals apart to be genuinely different options.

$L_1$'s cost grows linearly, so shrinking one feature's change to exactly zero is always worthwhile; $L_2$ penalises large deviations quadratically, so it "prefers" many small nonzero changes over one big one — never quite hitting zero.

A hard equality on a discrete label gives no gradient to follow. Hinge on the probability is smooth and non-zero until we cross $0.5$, providing a direction for gradient descent to push.

The minimal pixels it must change to flip the label are the pixels the model most relies on. If tiny ear-tweaks flip cat→dog, the model leaned on ear shape — a debugging signal, see Model Debugging.

We want the nearest counterfactual. Starting at $\mathbf{x}$ biases gradient descent to stay close and cross the boundary at the cheapest spot instead of wandering to a far-off valid region.

Validity only checks the flipped label. Two edits with the same $L_1$ cost can differ in actionability (one changes a mutable feature, one an immutable one) and in causal side-effects — worth lives in those extra dimensions.

A large-but-finite weight still lets the optimiser touch the feature if flipping is desperate enough. Infinity makes any change infinitely costly, guaranteeing it never moves.

There is none to find — the request is degenerate. A well-built system returns "no change needed," since the minimal edit is the zero edit.

Then no valid recourse exists. The honest output is "no achievable counterfactual," which itself is important information about model unfairness — see Recourse.

It belongs to neither class cleanly; an infinitesimal push either way flips it. The hinge is exactly zero at $p=0.5$, so proximity dominates and the counterfactual is essentially $\mathbf{x}$ itself.

The $\lambda$ weight on the prediction term was too small relative to proximity, so the search stopped before crossing. Increase $\lambda$ (or run more steps) — validity was never achieved.

A plain clip to $[\text{lower},\text{upper}]$ isn't enough; we must project onto $x_i^{\prime }\ge x_i$ after each step. This is a directional constraint that a causal/FACE method encodes explicitly.

Editing one should move the other. A method blind to the Causal Inference structure produces an impossible combo (tenure up, income magically flat); a DAG-constrained method updates the descendant automatically.

The prediction term dominates entirely; the search drives $p$ far past $0.5$ regardless of distance, often landing on an implausible, adversarial-looking point — the exact failure the plausibility term $\gamma$ exists to prevent.