ListNet Listwise Ranking Loss

Background

ListNet treats learning-to-rank as a probability distribution problem: define a softmax-derived distribution over permutations from both the predicted scores and the true relevance labels, then minimise their cross-entropy. The simplest listwise loss in production use.

Problem statement

Implement listnet_loss(scores, relevances). Let $p^{\text{pred}} = \text{softmax}(\text{scores})$ and $p^{\text{true}} = \text{softmax}(\text{relevances})$ over the candidate list. The loss is the cross-entropy:

$$\mathcal L \;=\; -\sum_i p^{\text{true}}_i \log p^{\text{pred}}_i$$

Use numerically stable softmax (subtract max before exp).

Input

• scores — 1-D array of floats, length $n$, the model's predicted scores.
• relevances — 1-D array of floats, length $n$, the true relevance labels.

Output

Returns a Python float >= 0.

Examples

Example 1 — identical scores and relevances → low loss

Input:  scores = relevances = [1.0, 2.0, 3.0]
Output: ≈ entropy of softmax([1,2,3])

Example 2 — reversed scores → higher loss

Input:  scores = reversed(relevances)
Output: much higher than the matched case

Constraints

• Length mismatch raises ValueError.
• Softmax must be numerically stable (subtract max).
• Returns a plain Python float.

Notes

• Top-1 vs top-k. This is the top-1 ListNet (probability the item is the best). The original paper extends to top-$k$ permutations; top-1 is the version that scales.
• vs pairwise. Listwise sees all items at once and balances them; pairwise (BPR / RankNet) sees one pair at a time. Listwise generally gives a small NDCG improvement at scale.