I think that this means that certain rank-based metrics, e.g. NDCG or AUC, which are very commonly used in ranking or classification, can be represented in a di

I think that this means that certain rank-based metrics, e.g. NDCG or AUC, which are very commonly used in ranking or classification, can be represented in a differentiable form. That, practically, means that neural networks can easily optimize directly for those results, instead of (most commonly) minimizing log-likelihood and monitoring for results on NDCG (as in the original formulation of RankNet). My (limited) understanding of LambdaRank is that their model empirically minimizes NDCG, but does not have a strong theoretical backing for why it should work. The experiments section is pretty clear in what potential applications can be, e.g. "optimizing directly for top-k classification loss" or "label ranking via soft Spearman’s rank correlation coefficient". In Google's case, there are pretty clear applications towards web search (top-k results, classification = do you click or not), and things like entity labeling (e.g. what label should we assign to a news story).