Learning to Rank

Learning to Rank formulates "score and order the candidate set for a query" as supervised learning in three paradigms: Pointwise regresses/classifies each document independently (e.g. CTR prediction) with no regard for relative order; Pairwise converts ranking into document-pair binary classification (document $i$ should rank above $j$), e.g. RankNet and RankSVM, covering only local pairwise relations; Listwise optimizes a whole-list ranking metric directly, e.g. ListNet and LambdaRank. LambdaMART = GBDT + LambdaRank is the de-facto standard for tabular ranking. For each pair $(i, j)$ it defines the Lambda gradient: $\lambda_{ij} = -\frac{\sigma \Delta NDCG}{1 + e^{\sigma(s_i - s_j)}}$, where $s_i, s_j$ are current scores, $\Delta NDCG$ is the NDCG change from swapping $i$ and $j$, and $\sigma$ controls the sigmoid steepness; the negative sign moves toward improving NDCG — a non-differentiable ranking metric weighting a differentiable pairwise gradient. Each document accumulates $\lambda_i = \sum_j \lambda_{ij}$, which the regression trees directly fit.

the core of search, recommendation and ad ranking; interview questions like "compare Pointwise/Pairwise/Listwise" or "where does LambdaMART's λ come from"; in practice, LightGBM/XGBoost rank objectives (lambdarank) train directly and position-sensitive metrics such as NDCG/MRR evaluate the result.

Pointwise optimizes per-item error and ignores order (errors near the top cost more); Pairwise only handles pairwise relations and treats "1→2" and "99→100" equally; Listwise/LambdaMART uses $\Delta NDCG$ so the training objective matches the online metric — the higher the position and the more relevant the document, the larger the gradient weight — directly optimizing the non-differentiable NDCG.

Compare Pointwise / Pairwise / Listwise: objective function, pros/cons and representative algorithms for each?

Write the LambdaMART gradient $\lambda_{ij} = -\frac{\sigma \Delta NDCG}{1 + e^{\sigma(s_i - s_j)}}$; explain the roles of $\Delta NDCG$ and $\sigma$, and why the negative sign?

Why does LambdaMART turn the non-differentiable NDCG into a differentiable regression target? What exactly does each GBDT tree fit?

How is LambdaRank's λ derived from RankNet's pairwise loss gradient?

Why use NDCG/MRR/GAUC instead of AUC for ranking? What is the essential difference from classification metrics?