Learning to Approximate a Bregman Divergence

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error $O_p(m^{-1/2})$ for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.Comment: 19 pages, 4 figure

Probabilistic Models over Ordered Partitions with Application in Learning to Rank

This paper addresses the general problem of modelling and learning rank data with ties. We propose a probabilistic generative model, that models the process as permutations over partitions. This results in super-exponential combinatorial state space with unknown numbers of partitions and unknown ordering among them. We approach the problem from the discrete choice theory, where subsets are chosen in a stagewise manner, reducing the state space per each stage significantly. Further, we show that with suitable parameterisation, we can still learn the models in linear time. We evaluate the proposed models on the problem of learning to rank with the data from the recently held Yahoo! challenge, and demonstrate that the models are competitive against well-known rivals.Comment: 19 pages, 2 figure

Active Sampling of Pairs and Points for Large-scale Linear Bipartite Ranking

Bipartite ranking is a fundamental ranking problem that learns to order relevant instances ahead of irrelevant ones. The pair-wise approach for bi-partite ranking construct a quadratic number of pairs to solve the problem, which is infeasible for large-scale data sets. The point-wise approach, albeit more efficient, often results in inferior performance. That is, it is difficult to conduct bipartite ranking accurately and efficiently at the same time. In this paper, we develop a novel active sampling scheme within the pair-wise approach to conduct bipartite ranking efficiently. The scheme is inspired from active learning and can reach a competitive ranking performance while focusing only on a small subset of the many pairs during training. Moreover, we propose a general Combined Ranking and Classification (CRC) framework to accurately conduct bipartite ranking. The framework unifies point-wise and pair-wise approaches and is simply based on the idea of treating each instance point as a pseudo-pair. Experiments on 14 real-word large-scale data sets demonstrate that the proposed algorithm of Active Sampling within CRC, when coupled with a linear Support Vector Machine, usually outperforms state-of-the-art point-wise and pair-wise ranking approaches in terms of both accuracy and efficiency.Comment: a shorter version was presented in ACML 201

Functional Bipartite Ranking: a Wavelet-Based Filtering Approach

It is the main goal of this article to address the bipartite ranking issue from the perspective of functional data analysis (FDA). Given a training set of independent realizations of a (possibly sampled) second-order random function with a (locally) smooth autocorrelation structure and to which a binary label is randomly assigned, the objective is to learn a scoring function s with optimal ROC curve. Based on linear/nonlinear wavelet-based approximations, it is shown how to select compact finite dimensional representations of the input curves adaptively, in order to build accurate ranking rules, using recent advances in the ranking problem for multivariate data with binary feedback. Beyond theoretical considerations, the performance of the learning methods for functional bipartite ranking proposed in this paper are illustrated by numerical experiments