Pinball (Quantile) Regression Loss

Background

Quantile regression predicts a specific quantile of $y \mid x$ rather than the mean. The loss is the asymmetric pinball loss: under-prediction is penalised by $\tau$ per unit, over-prediction by $(1 - \tau)$ per unit. The ETA-prediction case uses this to train per-quantile heads (p10, p50, p90) for delivery-time confidence intervals.

Problem statement

Implement pinball_loss(y_true, y_pred, tau):

$$\mathcal{L}(y, \hat{y}; \tau) \;=\; \max\bigl(\tau (y - \hat{y}),\ (\tau - 1)(y - \hat{y})\bigr)$$

Return the mean over all samples.

Input

• y_true, y_pred - array-like of floats, same length.
• tau - float in $(0, 1)$, the target quantile.

Output

• float >= 0.

Examples

Input: y_true=[1.0, 2.0], y_pred=[0.0, 0.0], tau=0.5
Output: 0.5 * mean([1.0, 2.0]) = 0.75

Constraints

• $0 < \tau < 1$. ValueError otherwise.
• Lengths must match.

Notes

• tau = 0.5 -> MAE / 2. The 50%-quantile pinball loss equals half the mean absolute error.
• Asymmetric penalty. For ETA prediction with $\tau = 0.9$, under-promising (predicting too high) is 9x cheaper than missing the deadline (predicting too low).