Focal Loss for Imbalanced Data

When 99% of examples are easy negatives (background in object detection, legitimate transactions in fraud), their small losses still add up and swamp the rare positives. Focal loss down-weights easy examples:

where for a binary label $y$ and predicted probability $p$: $p_t = p$ if $y = 1$, else $1 - p$; and $\alpha_t = \alpha$ if $y = 1$, else $1 - \alpha$.

Implement focal_loss(logits, y, alpha=0.25, gamma=2.0) returning the mean loss. Compute $\log p_t$ stably from the logits: $\log p = -\text{softplus}(-z)$ and $\log(1-p) = -\text{softplus}(z)$, using np.logaddexp(0, ·). With gamma=0 and alpha=0.5 it equals half the binary cross-entropy.