Antithetic and Monte Carlo kernel estimators for partial rankings

TL;DR

This paper introduces novel Monte Carlo and antithetic kernel estimators for partial rankings, enabling more efficient and accurate analysis of incomplete ranking data in machine learning applications.

Contribution

It extends kernel methods for complete rankings to partial rankings using consistent Monte Carlo estimators and introduces an antithetic variance reduction scheme for improved performance.

Findings

Antithetic kernel estimator has lower variance.

Empirical results show better performance in ML tasks.

Provides a computationally tractable alternative for partial rankings.

Abstract

In the modern age, rankings data is ubiquitous and it is useful for a variety of applications such as recommender systems, multi-object tracking and preference learning. However, most rankings data encountered in the real world is incomplete, which prevents the direct application of existing modelling tools for complete rankings. Our contribution is a novel way to extend kernel methods for complete rankings to partial rankings, via consistent Monte Carlo estimators for Gram matrices: matrices of kernel values between pairs of observations. We also present a novel variance reduction scheme based on an antithetic variate construction between permutations to obtain an improved estimator for the Mallows kernel. The corresponding antithetic kernel estimator has lower variance and we demonstrate empirically that it has a better performance in a variety of Machine Learning tasks. Both kernel estimators are based on extending kernel mean embeddings to the embedding of a set of full rankings consistent with an observed partial ranking. They form a computationally tractable alternative to previous approaches for partial rankings data. An overview of the existing kernels and metrics for permutations is also provided.