For massive and heterogeneous modern datasets, it is of fundamental interest to provide guarantees on the accuracy of estimation when computational resources are limited. In the application of learning to rank, we provide a hierarchy of rank-breaking mechanisms ordered by the complexity in thus generated sketch of the data. This allows the number of data points collected to be gracefully traded off against computational resources available, while guaranteeing the desired level of accuracy. Theoretical guarantees on the proposed generalized rank-breaking implicitly provide such trade-offs, which can be explicitly characterized under certain canonical scenarios on the structure of the data.
@cite_10: 
@cite_9: Kemeny Rank Aggregation is a consensus finding problem important in many areas ranging from classical voting over web search and databases to bioinformatics. The underlying decision problem Kemeny Score is NP-complete even in case of four input rankings to be aggregated into a "median ranking". We analyze efficient polynomial-time data reduction rules with provable performance bounds that allow us to find even all optimal median rankings. We show that our reduced instances contain at most [InlineEquation not available: see fulltext.] candidates where @math d a denotes the average Kendall's tau distance between the input votes. On the theoretical side, this improves a corresponding result for a "partial problem kernel" from quadratic to linear size. In this context we provide a theoretical analysis of a commonly used data reduction. On the practical side, we provide experimental results with data based on web search and sport competitions, e.g., computing optimal median rankings for real-world instances with more than 100 candidates within milliseconds. Moreover, we perform experiments with randomly generated data based on two random distribution models for permutations.
@cite_1: There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.
However, in modern applications, the computational complexity of the existing approaches blow-up due to the heterogeneity of modern datasets. Although, statistical and computational tradeoffs have been investigated under other popular choice models such as the Mallows models by @cite_9 or stochastically transitive models by @cite_1 , the algorithmic solutions do not apply to random utility models and the analysis techniques do not extend. We provide a novel rank-breaking algorithms and provide finite sample complexity analysis under the PL model. This approach readily generalizes to some RUMs such as the flipped Gumbel distribution. However, it is also known from @cite_10 , that for general RUMs there is no consistent rank-breaking, and the proposed approach does not generalize.