Pairwise Ranking Loss (BPR / RankNet)

Background

Pairwise ranking loss treats ranking as a series of pairwise preference choices: positive item $i$ should outrank negative item $j$. The Bradley-Terry / RankNet form uses a logistic on the score difference; BPR (Bayesian Personalised Ranking) is the same loss applied to implicit-feedback recsys.

Problem statement

Implement pairwise_ranking_loss(scores_pos, scores_neg) returning the mean BPR/RankNet loss:

$$\mathcal{L} \;=\; -\frac{1}{N}\sum_i \ln \sigma\!\bigl(s^+_i - s^-_i\bigr)$$

where $s^+_i, s^-_i$ are the model scores for the positive and negative items in pair $i$.

Input

• scores_pos - 1-D array of positive scores, length $N$.
• scores_neg - 1-D array of negative scores, length $N$.

Output

• float >= 0, the mean loss.

Examples

Input: scores_pos=[2.0], scores_neg=[1.0]
Output: -ln(sigma(1)) ~= 0.3133

Constraints

• Use a numerically stable softplus: $-\ln \sigma(x) = \mathrm{softplus}(-x)$. The naive form overflows for very negative $x$.
• Lengths must match; ValueError otherwise.

Notes

• Pairwise vs pointwise. Pointwise (BCE on individual items) optimises calibration; pairwise optimises ordering. Production rankers often use pointwise + pairwise auxiliary.