Conformal Scores and Regions

Residual-based scores

• L2Score: \(s(x,y) = \|f(x) - y\|_2\) → L2 ball.
• L1Score: \(s(x,y) = \|f(x) - y\|_1\) → L1 ball.
• LinfScore: \(s(x,y) = \|f(x) - y\|_\infty\) → Linf ball.
• MahalanobisScore: \(s(x,y) = \sqrt{(f(x)-y)^\top W (f(x)-y)}\) → ellipsoid.

GPCPScore (union of balls)

• Score: \(s(x,y) = \min_{k} \| \text{sample}_k(x) - y \|_2\).
• Predictor returns K samples; prediction region becomes a union of L2 balls centered at the samples with radius equal to the conformal quantile.
• Useful with generative models that can sample conditional outputs.

Calibrators

• SplitConformalCalibrator runs split conformal quantile estimation given a predictor and score.
• For GPCPScore, predictor should return samples (shape K x batch x d); the score builds the union region.

Robust optimization paths

• Deterministic (affine in uncertainty): use robustify_affine_objective / robustify_affine_leq with region.support_function (unions use max of component supports).
• Scenario-based: ScenarioRobustOptimizer samples from regions (works for unions).
• Gradient-based Danskin: DanskinRobustOptimizer solves inner maximization per component (works for unions) and updates w via gradients (optionally autograd).
• Example (capacity planning): \(\min_{0 \le c \le \bar{c}} \max_{\lambda \in \mathcal{C}(x)} c_{\text{cap}} c + c_{\text{short}} (\lambda - \mu c)^+\) with \(\mathcal{C}(x)\) an L2 interval.

Region volume

• Regions provide volume when closed-form (L2/L1/Linf balls, ellipsoids); unions return None.
• Use region.volume_mc(bounds) to estimate volume for unions or complex regions.