A Theoretical Analysis of Normalized Discounted Cumulative Gain (NDCG) Ranking Measures

A central problem in ranking is to design a measure for evaluation of ranking functions. In this paper we study, from a theoretical perspective, the Normalized Discounted Cumulative Gain (NDCG) which is a family of ranking measures widely used in practice. Although there are extensive empirical studies of NDCG, little is known about its theoretical properties. We first show that, whatever the ranking function is, the standard NDCG which adopts a logarithmic discount, converges to 1 as the number of items to rank goes to infinity. On the first sight, this result seems to imply that the standard NDCG cannot differentiate good and bad ranking functions on large datasets, contradicting to its empirical success in many applications. In order to have a deeper understanding of the general NDCG ranking measures, we propose a notion referred to as consistent distinguishability. This notion captures the intuition that a ranking measure should have such a property: For every pair of substantially different ranking functions, the ranking measure can decide which one is better in a consistent manner on almost all datasets. We show that the standard NDCG has consistent distinguishability although it converges to the same limit for all ranking functions. We next characterize the set of all feasible discount functions for NDCG according to the concept of consistent distinguishability. Specifically we show that whether an NDCG measure has consistent distinguishability depends on how fast the discount decays; and r−1 is a critical point. We then turn to the cut-off version of NDCG, i.e., NDCG@k. We analyze the distinguishability of NDCG@k for various choices of k and the discount functions. Experimental results on real Web search datasets agree well with the theory.